SMITHSONIAN MISCELLANEOUS COLLECTIONSVolume 74^ Number 1 SMITHSONIAN MATHEMATICAL FORMULAEANDTABLES OF ELLIPTIC FUNCTIONSMathematical Formulae Prepared byEDWIN P. ADAMS, Ph.D.PROFESSOR OF PHYSICS, PRINCETON UNIVERSITYTables of Elliptic Functions Prepared under the Direction ofSir George Greenhill, Bart.COL. R. L.' HIPPISLEY, C.B. Publication 2672 CITY OF WASHINGTONPUBLISHED BY THE SMITHSONIAN INSTITUTION1922 ADVERTISEMENTThe Smithsonian Institution has maintained for many years a group ofpublications in the nature of handy books of information on geographical,meteorological, physical, and mathematical subjects. These include theSmithsonian Geographical Tables (third edition, reprint, 1918); the SmithsonianMeteorological Tables (fourth revised edition, 1918); the Smithsonian PhysicalTables (seventh revised edition, 1921); and the Smithsonian MathematicalTables: Hyperbolic Functions (second reprint, 1921).The present volume comprises the most important formulae of many branchesof applied mathematics, an illustrated discussion of the methods of mechanicalintegration, and tables of elliptic functions. The volume has been compiled byDr. E. P. Adams, of Princeton University. Prof. F. R. Moulton, of the Univer-sity of Chicago, contributed the section on numerical solution of differentialequations. The tables of elliptic functions were prepared by Col. R. L. Hippisley,C. B., under the direction of Sir George Greenhill, Bart., who has contributed theintroduction to these tables.The compiler. Dr. Adams, and the Smithsonian Institution are indebted tomany physicists and mathematicians, especially to Dr. IJ. L. Curtis and col-leagues of the Bureau of Standards, for advice, criticism, and cooperation inthe preparation of this volume. Charles D. Walcott,Secretary of the Smithsonian Institution.May, ig22. PREFACEThe original object of this collection of mathematical formulae was to bringtogether, compactly, some of the more useful results of mathematical analysisfor the benefit of those who regard mathematics as a tool, and not as an end initself. There are many such results that are difficult to remember, for one whois not constantly using them, and to find them one is obliged to look through anumber of books which may not immediately be accessible.A collection of formulae, to meet the object of the present one, must belargely a matter of individual selection; for this reason this volume is issuedin an interleaved edition, so that additions, meeting individual needs, may bemade, and be readily available for reference.It was not originally intended to include any tables of functions in thisvolume, but merely to give references to such tables. An exception was made,however, in favor of the tables of elliptic functions, calculated, on Sir GeorgeGreenhill's new plan, by Colonel Hippisley, which were fortunately secured forthis volume, inasmuch as these tables are not otherwise available.In order to keep the volume within reasonable bounds, no tables of indefiniteand definite integrals have been included. For a brief collection, that of thelate Professor B. O.'Peirce can hardly be improved upon; and the elaboratecollection of definite integrals by Bierens de Haan show how inadequate anybrief tables of definite integrals would be. A short list of useful tables of thiskind, as well as of other volumes, having an object similar to this one, is appended.Should the plan of this collection meet with favor, it is hoped that suggestionsfor improving it and making it more generally useful may be received.To Professor Moulton, for contributing the chapter on the NumericalIntegration of Differential Equations, and to Sir George Greenhill, for his Intro-duction to the Tables of Elliptic Functions, I wish to express my gratitude.And I wish also to record my obligations to the Secretary of the Smithsonian In-stitution, and to Dr. C. G. Abbot, Assistant Secretary of the Institution, for theway in which they have met all my suggestions with regard to this volume.E. P. AdamsPrinceton, New Jersey COLLECTIONS OF MATHEMATICAL FORMULAE, ETC.B. O. Peirce: A Short Table of Integrals, Boston, 1899.G. Petit Bois: Tables d'lntegrales Indefinies, Paris, 1906.T. J. I'A. Bromwich: Elementary Integrals, Cambridge, 191 1.D. Bierens de Ha.a.n: Nouvelles Tables dTntegrales Definies, Leiden, 1867.E. Jahnke and F. Emde: Funktionentafeln mit Formeln und Kurven, Leipzig,1909.G. S. Carr: a Synopsis of Elementary Results in Pure and Applied Mathe-matics, London, 1880.W. Laska: Sammlung von Formeln der reinen und angewandten Mathematik,Braunschweig, 1 888-1 894.W. LiGOWSKi: Taschenbuch der Mathematik, Berlin, 1893.O. Th. Burklen: Formelsammlung und Repetitorium der Mathematik,Berlin, 1922.F. Auerbach: Taschenbuch fur Mathematiker und Physiker, i. Jahrgang,1909. Leipzig, 1909. CONTENTS PAGESymbols viiiI. Algebra iII. Geometry 29III. Trigonometry 61IV. Vector Analysis 91V. Curvilinear Coordinates 99VI. Infinite Series 109VII. Special Applications of Analysis 145VIII. Differential Equations 162IX. Differential Equations (Continued) 191X. Numerical Solution of Differential Equations 220XI. Elliptic Functions 243Introduction by Sir George Greenhill, F.R.S 245Tables of Elliptic Functions, by Col. R. L. Hippisley .... 259Index 311 vii SYMBOLSlog logarithm. Whenever used the Naperian "logarithm is understood.To find the common logarithm to base lo : logio a = 0.43429 . . • log a.log a = 2.30259 . . . logio a. ! Factorial, nl where n is an integer denotes 1.2.3.4Equivalent notation ^4: Does not equal.> Greater than.< Less than.^ Greater than, or equal to.:^ Less than, or equal to.Binomial coefficient. See 1.51-CO — ^ Approaches. I flffc I Determinant where aik is the element in the iih. row and ^th column, , ^'—^' ' ' ' \ Functional determinant. See 1.37.d{Xi, X2 ) I a 1 Absolute value of a. If a is a real quantity its "numerical value,without regard to sign. If a is a complex quantity, a = a + i^, \ a] = modulus of a = +Va^ + jS^,i The imaginary = +V — i.^ Sign of summation, i.e., ^dk = ai + 02 + 03 + • . • • + On.k=nII Product, i.e., I I (i + kx) = (i + x){i + 2x){i + 3^) . . . . (i + nx). vui I. ALGEBRA1.00 Algebraic Identities.1. a" - b" = (a - b){a"--^ + a^'-b + a'^'W + + ab""-- + b""'^).2. a" ± Z>" = (a + 6)(a"-i - a"-'^b + a"-^^- - =F ab"-'- ± fi''-^.« odd: upper sign.n even: lower sign.3. (a; + ai){x + 0-2) (x- + Qn) = X" + Pix"-i + P2a;"-2 ++ Pn-lX + Pn.Pi = Ol + 02 + +an,Pk = sum of all the products of the a's taken ^ at a time.P„ = aiaoQs . . . . o„.(a2 + &2)(a2 + i82) = (aa T 6/3)2 + (a (3 ± ^>a)2.(a2 - 62)(a2 _ ^2) ^ (^ct ^ J^)2 _ (^^ ^ ^^)2,(^2 + 52 + ,2)(a2 + /32 _|. y) = (^c^ + ^^ + ^yy ^ Q^y _ ^,)2 + (,a _ ^^)2+ (aiS - a6)2.(a2 + b' + c2 + ^^)(a2 + |82 + y + 52) ^ (^ct + 6/3 + C7 + rfS)^+ (fljS - ba + c5 - dyf + (^7 - 66 - ca + d^Y + (a5 + 67 - CjS - c?a)2.{ac - bdY + (ai + bcY = {ac + bdY + {ad - be)-,(a + b)(b + c)(c + a) = (a + b + c){ab + be + ca) -abc.(a + b){b + c){c + a) = a^{b + c) + b"{c + a) + c\a + b) + 2abc.(a + b)(b + c){c + a) = bc(b + c) + ca{c + a) + ab{a + 6) + 2abc.3(a + b){b + c){c + a) = {a + b + cY- {a' + b' + c').(b - a){c - a){c - b) = a\c - b) + b~(a - c) + c-{b - a).(6 - a)(c - a)(c - b) = a{F- - c") + 6(^2 - a") + cia" - 6^).(6 - a){c - a){c - b) = bc{c - b) + ca{a - c) + ab{b - a).(a - bY +{b- cY + (c - aY = 2[(a - b){a - c) + {b - a){b - c)+ (c- a)(c- 6)].a3(52 _ ^2) _^ ^3(^2 _ ^2) ^ ^3(^2 _ ^2) = (^ _ J) (^ _ ^ ) (^ _ (-)(a^ + 6c + CO) . (o + 6 + c)(a2 + 62 + c2) = 6c(6 + c) + ca(c + o) + ab{a + 6) + a^ + 6^ + c^.(a + 6 + c)(6c + CO + a6) = 0^(6 + c) + b'{c + a) + c2(a -\- b) + 7,abc.{b + c- d){c + a - b){a + b - c) = a^ib + c) + b\c + c) + c2(a + 6) -(a3 + 63 + c3 + 2o6c). 2 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS21. {a + b + c){ - a + b + c){a - b + c){a ^ b - c) = 2{b''c^ + cV + a^i^) -{a' + b' + c*).22. {a + b + c + dy + (a + b - c - dy + {a + c - b - d)- + {a + d - b - cY= 4(a2 + ^2 + C-' + d^).If /I = ca + 67 + Cj8B = a^ + ba + cyC = ay + b^ + ca23. (a + 6 + c)(a + (3 + y)=A+B + C.24. [a^ + 62 + c2 - (a6 + bc + ca)'] [a^ + /^^ + y^ - {a(3 + I3y + 7a)] 25. (a3 + 63 ^ c* - 30k) (a^ + j8^ + 7^ - 3 a/37) = ^^ + 5'^ + - 3ABC.ALGEBRAIC EQUATIONS1.200 The expressionf(x) = co-v" + aix"~i + 02'^"~^ + + an-ix + anis an integral rational function, or a polynomial, of the ;zth degree in x.1.201 The equation f{x) = o has n roots which may be real or complex, dis-tinct or repeated.1.202 If the roots of the equation f(x) = o are Ci, C2, . . ., Cn,f(x) = ao(x - Ci){x - C2) {x - Cn)1.203 Symmetric functions of the roots are expressions giving certain com-binations of the roots in terms of the coefficients. Among the more importantare : Ci + C2-\- + Cn = ao a<2,C1C2 + C1C3 + . . . + C2C3 + CoCi+ + Cn-iCn = — C1C2C3 + CiC2Ci + . . . + Cidd + + Cn-2Cn-lCn = ao C\C2Cz C„ = (-1)"— • Oo1.204 Newton's Theorem. If si, denotes the sum of the ^th powers of all theroots of iix) = o, Sk == Ci + C2 + + CnIfll + 51^0 = O2^2 + SiOi + 5ocro = '303 + Sia2 + 52^1 + ssao = o4O4 + SiOs + S2ai -\- SiOi + 54^0 = o •^3 = - -— 4 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS1.221 It follows from 1.220 that J\h) = R. This gives a convenient way ofevaluating J{x) for x = h.1.222 To express J{x) in the form : J{x) = Ao{x - hy + A,{x - hy-^ + .... + An-i{x -h)+ An.By 1.220 form J{h) = An. Repeat this process with each quotient, and thelast term of each line of sums will be a succeeding value of the series of co-efficients An, An-1, , Aq.Example : f{x) = 2,X^ + 2X'* - SX" + 2X - 4 h = 22 O -4 _6 i6 32 48 1008 16 24 50 96 = As6 28 88 22414 ALGEBRA 5the upper sign being used if the roots are to be diminished and the lower signif they are to be increased. The resulting equation will be : j{±h) + .v/(±/o + ^'r (±//) + j-;/'"(±//) +— =where /'(.v) is the first derivative of /(.v), J"{x), the second derivative, etc.The resulting equation may also be written : y4o.v" + .4i.v"-i + A.2X"-~ + + An-lX -^ An = owhere the coefficients may be found by the method of 1.222 if the roots are tobe diminished. To increase the roots by h change the sign of h. MULTIPLE ROOTS1.240 If c is a multiple root of J{x) = o, of order m, i.e.. repeated m times,then fix) = (x - c)-Q; R = oc is also a multiple root of order m - i oi the first derived equation, f'(x) = o;of order m — 2 of the second derived equation, f'{x) = o, and so on.1.241 The equation /(x) = o will have no multiple roots ii fix) and/(x) haveno common divisor. If F{x) is the greatest common divisor oi f{x) and /'(a;),f{x)/F{x) =/iGv), and/i(x) will have no multiple roots. 1.250 An equation of odd degree, n, has at least one real root whose sign isopposite to that of a„.1.251 An equation of even degree, n, has one positive and one negative realroot if a„ is negative.1.252 The equation f(x) = o has as many real roots between x = Xl and x = X2as there are changes of sign in f(x) between Xi and X2.1.253 Descartes' Rule of Signs: No equation can have more positive rootsthan it has changes of sign from + to - and from - to +, in the terms of /(.v).No equation can have more negative roots than there are changes of sign in f{-x).1.254 If f{x) = o is put in the form^oCv - h)- + Ai{x - h)"-^ + + An = oby 1.222, and Ao, Ai, , An are all positive, h is an upper limit of thepositive roots.If /(-a-) = o is put in a similar form, and the coefiicients are all positive,h is a lower limit of the negative roots. O MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSIf /(i/x) = o is put in a similar form, and the coefficients are all positive,A is a lower limit of the positive roots. And with /(- i/x) = o, h is an upperlimit of the negative roots.1.255 Sturm's Theorem. Form the functions:fix) = oox" + aix-"~i + a-iX"-' + . . . . + a„/lOv) =/(x) = naox""-^ + (n - i)aiX"-2 + . , . . + a^_jMx) = -R, in fix) = Q^Mx) + R,Mx) = -R2 in Mx) = Q2f2ix) + R2 The number of real roots of f{x) = o between x = Xi and x = ^2 is equal to thenumber of changes of sign in the series f(x), fiix), Mx), . . . when Xi is sub-stituted for x minus the number of changes of sign in the same series when X2is substituted for x. In forming the functions /i, /2, . . . . numerical factorsmay be introduced or suppressed in order to remove fractional coefficients.Example : /(x) = X* - 2x^ - 3X- + Iox - 4/i(x) = 2X^ - 3x2 _ 2.V + 5/2(x) = 9X2 _ 27.V+ II/3(x) = -8x - 3/4(X) = -1433 ALGEBRAForm a third row by cross-multiplication: ,n-2 Oia2 — 00O3 diOA — Oo(i5 dide — Clod?ai ai OiForm a fourth row by operating on these last two rows by a similar cross-multiplication. Continue this operation until there are no terms left. Thenumber of variations of sign in the first column gives the number of rootswhose real parts are positive.If there are any equal roots some of the subsidiary functions will vanish.In place of one which vanishes write the differential coeflEicient of the last onewhich does not vanish and proceed in the same way. At the left of each rowis written the power of x corresponding to the first subsidiary function in thatrow. This power diminishes by 2 for each succeeding coefficient in the row.Any row may be multiplied or divided by any positive quantity in orderto remove fractions. DETERMINATION OF THE ROOTS OF AN EQUATION1.260 Newton's Method. If a root of the equation f(x) = o is known to liebetween Xi and 0C2 its value can be found to any desired degree of approximationby Newton's method. This method can be applied to transcendental equationsas well as to algebraic equations.If b is an approximate value of a root,b — yjT: = c is a second approximation, c — -p-r\ = c^ is a third approximation.This process may be repeated indefinitely.1.261 Horner's Method for approximating to the real roots of f{x) = o.Let pi be the first approximation, such that pi -\- x > c > pi, where c is theroot sought. The equation can always be transformed into one in which thiscondition holds by multiplying or dividing the roots by some power of 10by 1.231. Diminish the roots by pi by 1.233. In the transformed equationA(i{x - px)^ + Ai{x - PiY-^ + ....+ An-i{x - pi) +An = oput h - ^"10 ~ An-land diminish the roots by />2/io, yielding a second transformed equationbIx -Pi- ^y + bJx -Pi- ^)«-i + . . . . + Bn = o.\ 10/ V 10/ 8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSIf Bn and Bn-i are of the same sign p2 was taken too large and must be dimin-ished. Then take pz _ BnlOO Bn-1and continue the operation. The required root will be: ^ lO lOO1.262 Graeffe's Method. This method determines approximate values of allthe roots of a numerical equation, complex as well as real. Write the equationof the «th degreef{x) = ao.v"" — ai.v"~^ + a^x""^ — . . . . ± a„ = o.The product contains only even powers of x. It is an equation of the wth degree in x"^. Thecoefficients are determined byAo = ao2Ai = ai^ — 2a (jO^A% = a^ — 201(73 + 200^4Az = az' — 202^4 + 20105 — 200^6yl4 = a^ — 203O5 + 202O6 - 201O7 + 200O8 The roots of the equationA^y"" - ^ly"-^ + Aiy""-^ - . . . . ± ^„ = oare the squares of the roots of the given equation. Continuing this process weget an equation i?0M" - i?lM"~^ + i?2M"~^ - . . . . ± /?„ = Owhose roots are the 2'"th powers of the roots of the given equation. Put X = 2^Let the roots of the given equation be Ci, c^, . . . . , Cn. Suppose first thatCi > C2 > Cs > > CnThen for large values of X, If the roots are real they may be determined by extracting the Xth roots ofthese quantities. Whether they are ± is determined by taking the sign whichapproximately satisfies the equation f(x) = o.Suppose next that complex roots enter so that there are equalities amongthe absolute values of the roots. Suppose that I ci I ^ I C2 I ^ 1 rs I ^ ^ I 0> I ; I Cp I > I Cp+i 1 ; I Cp+i 1 ^ I Cp+2 I ^ . . . . ^ I c„ 1 ALGEBRAThen if X is large enough so that Cp^ is large compared to Cp^^, c-^, c-^Cp^ approximately satisfy the equation:RouP - RiuP-^ + RiUP-^ - . . . . ± i^p = o :„^ approximately satisfy the equation:RnU"' Rp+^U^^-P-^ + i?p+2M»-^-2 _ ^ ±RnTherefore when X is large enough the given equation breaks down into a numberof simpler equations. This stage is shown in the process of deriving the suc-cessive equations when certain of the coefficients are obtained from those ofthe preceding equation simply by squaring.References: Encyklopadie der Math. Wiss. I, i, 3a (Runge).Bairstow: Applied Aerodynamics, pp. 553-560; the solution of a numericalequation of the 8th degree is given by Graeffe's Method. 1.270 Quadratic Equations.The roots are : If 1.271 Cubic equations.Substitute where X- + 2ax + 6 = 0. Xi = -a + Va^ - bXi = —a — -s/a? — boci -\- xi = — 2aX1X2 = b.a? > b roots are real,a^ < b roots are complex,a? = b roots are equal.(i) x^ + ax^ + te + c = o. (2) x = y --(3) / - ?>py - 2q =3, = --.ab 2 ,20 = a^ — c.3 27Roots of (3) : li p > o, g > o, q^ > p^cosh Vp^ 10 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS yi = 2\/^ cosh —y2= -ys= -li p > o, q < o, q" > /»^cosh 4> = — + W^P sinh — - - ^VIp sinh - • li p o, q^ < p^ y2= - — + iV - 3P cosh —Ti . / 13^3 = --— tV- 3p cosh—2 3cos = —~Vp'yi = 2\/p cos y2 = ^ + V3p sin^ -vi /—- . 4)2 --• 31.272 Biquadratic equations.a^x^ + aix^ + a«::t;2 + os-'v + ^4 = 0.Substitute X = yy^^—^Ey-'+^Gy-^—.F^odo- ao"^ ao* ALGEBRAH = anQ^ - a^G = ao^az - 3aoai02 + 2ai^F = ao^ai - 4ao^Oifl3 + taoa^a^ — 3^1*/ = a 0^4 - 401^3 + 302^F = a^H - sH'J = aoOia-i + 2^10203 — ^003- — ai~ai — oz^A = /^ — 27/^ = the discriminantG2 + 4H' = ao'iHI - aoJ).Nature of the roots of the biquadratic:A = o Equal roots are presentTwo roots only equal : / and / are not both zeroThree roots are equal : I = J = oTwo distinct pairs of equal roots: G = o; ao'I — i2H^ =Four roots equal : H = I = J = o.A < o Two real and two complex rootsA > o Roots are either all real or all complex : H < o and atfl — 12^- < o Roots all realH > o and ao^I — i2H^ > o Roots all complex. DETERMINANTS1.300 A determinant of the «th order, with n- elements, is writtenan an an ai„a-2i 0-22 a^z a^nflsi ^32 '2dv,dxjl d{xi, X2, Jn);Xn) is the Jacobian,1.371 If y\, 3'2, , >'n are the partial derivatives of a functionF{xi, Xi, , Xn) : dFJi = -Q^_ {i = I, 2, . . . ., w)the symmetrical determinant:H dXi dXj JdF aF dF\\dxi' dxo * • • • QxJa(Xl, X2, , Xn)is the Hessian.1.372 If vi, A'2, , yn are given as implicit functions of .Ti, X2, ,Xn by the n equations : Fiiyi, y2, , yn, Xi, x^, , Xn) = o Fniyu 3'2, yn, Xl, Xo, . ., Xn) = Othen dhuvi, ,yn) ^ (_ y d(FuF2 ,F„) ^ d(Fr, F.2, . ..,F„)d(xu X2, . . ., Xn) ^ d{xx, xo, . . ., .T„) d(yu >'., . . . , t„)1.373 If the n functions yi, y^, , yn are not independent of each otherthe Jacobian, /, vanishes; and if 7 = o the w functions yi, >'2, . . . ., 3'„ are notindependent of each other but are connected by a relationF<'2, , >'n) = O ALGEBRA I 71.374 Covariant property. If the variables Xi, X2, . . . ., Xn are transformedby a linear substitution : Xi = an ^1 + OisS + + ain^r^ (i = I, 2, . . . ., n)and the functions yi, j2, , yn of Xi, :V2, , Xn become the functionsrji, r]2, , Vn of ^1, ^2, , ^n:w d(Vi, V2, , Vn) djy,, yo, , yn) , ,•^ ' a(|l, I2, , U dix,, X2, , Xn) ' *' 'or r = J-\ an 1where | a^ \ is the determinant or modulus of the transformation.For the Hessian, W = H'\ an 1 K 1.380 To change the variables in a multiple integral : / = / fFiyi, }'2, , yn)dyidy2 Jy„to new variables, xi, X2, . . . ., Xn when yi, ya, , yn are given functionsof Xi, Xi, , Xn'.i=r f ^^y^' y" > y^l F{x)dx,dx, dx.J J d{Xi, Xi, , Xn)where F{x) is the result of substituting X\, X2, . . . ., Xn for yi, y^, . . ., ynin /^(yi, y2, , yj. PERMUTATIONS AND COMBINATIONS1.400 Given n different elements. Represent each by a number, i, 2, 3, ,n. The number of permutations of the n different elements is, „Pn = n\e.g., w = 3 : (123), (132), (213), (231), (312), (321) =6 = 3!1.401 Given n different elements. The number of permutations in groups ofr {r"-^/)" - I/> - IIf /> + i) + 3(/? + 2) + + w(/> + w - l) = -w(w + l)(3^ + 2W - 2).5- i'-? + (/» - i) (9 - i) + (/> - 2) (g - 2) + (/> - w) (g - «) = ^^^[^/'g - (w - i) (3/» + 39 - 2w + i)]. 6. I I ^ I K^ + i) I ^ 5(6 + i). . . . (6 + w-i)o c(a +1) • • • • ^i^^ + i) . . . . (a + ;/ - i) ^ ^)(6 + i) . . . (^> + w) g - I(6 + I — a)o(o + i) . . . . (a + w — i) 6 + I — a II. GEOMETRY2.00 Transformation of coordinates in a plane.2.001 Change of origin. Let x, y he a, system of rectangular or oblique coor-dinates with origin at 0. Referred to x, y the coordinates of the new origin O'are a, b. Then referred to a parallel system of coordinates with origin at 0'the coordinates are x', y'. X = x' + ay = y' + b.2.002 Origin unchanged. Directions of axes changed. ObUque coordinates.Let OJ be the angle between the x - y axes measured counter-clockwise fromthe X- to the y-axis. Let the a;'-axis make an angle a with the x-axis and the/-axis an angle j8 with the x-axis. All angles are measured counter-clockwisefrom the x-axis. Then :» sin CO = x' sin {co - a) + y' sin (o) - /3)y sin CO = x' sin o: -J- y' sin /3co' = i8 - a.2.003 Rectangular axes. Let both new and old axes be rectangular, the newaxes being turned through an angle d with respect to the old axes. Then2 2 X = x' COS B — y' sin dy = x' sin d -\- y' cos d. 2.010 Polar coordinates. Let the y-axis make an angle co with the a;-axis andlet the X-axis be the initial line for a system of polar coordinates r, B. All anglesare measured in a counter-clockwise direction from the a;-axis. _ r sin (co - B)sin COsin By = r -.sm CO2.011 If the X, y axes are rectangular, co = — , X = r cos By = r sin B.29 30 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 2.020 Transformation of coordinates in three dimensions.2.021 Change of origin. Let x, y, z be a system of rectangular or oblique coor-dinates with origin at O. Referred to x, y, z the coordinates of the new origin0' are a, b, c. Then referred to a parallel system of coordinates with origin atO' the coordinates are x' , y' , z'. X = x' + ay = y + bz = z' + c2.022 Transformation from one to another rectangular system. Origin un-changed. The two systems are x, y, z and x' y' z' . Referred to .t, y, z the direction cosines of x' are /i, Wi, wiReferred to x, y, z the direction cosines of y' are h, m^, fhReferred to x, y, z the direction cosines of z' are h, nts, UsThe two systems are connected by the scheme : GEOMETRYCOS- B 31cos^ 7ffi2 + m - h - I «3 + k - nh - I /i + W2 - fii - I2.024 Transformation from a rectangular to an oblique system, x, y,tangular system: x\ y' , z' oblique system.cos xy' = hcos yy' = m2cos zy' = iHcos xx' = hcos yx' = micos zx' = til cos xz = Izcos ys' = Mscos 22 ' = W3X = /iX' + ^2)'' + IsZ'y = mix' + nhy' + Wsz'z = nix' + tiiy' + W32'cos y'z' = hh + W2W3 + ;Z2W3cos 2'x' = hh + W3W1 + nmicos x')'' = ^1/2 + minh + Wi'«^/i^ + m-^ + fix' = 1li + m^ + n^ = Ih^ + W3^ + W3^ = ITransformation from one to another oblique system.cos xx' = h 32 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS2.026 Transformation from one to another oblique system.If Hx, fly, Hz are the normals to the olanes yz, zx, xy and Wx', Uy', n/ thenormals to the planes y'z', z'x', x'y',X cos xux = x' cos x'nx + y' cos y'nx + z' cos z'nx.y cos yriy = x' cos x'ny + y' cos y'uy + z' cos z'uy.z cos zwz = x' cos .r'wz + y' cos )''wz + z' cos z'fiz. x' cos .t'«x' = X cos xwx' + y cos >'«i' + z cos znj .y' cos y'w y = X cos xw v' + >» cos yuy -{- z cos zw j,'.z' cos z'uz = X cos xnz' + y cos jWz' + z cos zwz'. 2.030 Transformation from rectangular to spherical polar coordinates.r, the radius vector to a point makes an angle 6 with the z-axis, the projectionof r on the x-y plane makes an angle with the x-axis.x = r sin 6 cos (j> r"^ = x"^ + ^ + z^y = r sin 6 sin cj) B = cos~^ =V -v^ + >'" + z^z = r cos 6 . , y(b = tan-i -a:2.031 Transformation from rectangular to cylindrical coordinates.p, the perpendicular from the z-axis to a point makes an angle 6 with thex-z plane. X = p cos p = V.T- + y^ Iy = p sin = tan-i ^2.032 Curvilinear coordinates in general.See 4.0 2.040 Eulerian Angles.Oxyz and Ox'y'z' are two systems of rectangular axes with the same origin 0.OK is perpendicular to the plane zOz' drawn so that if Oz is vertical, and theprojection of Oz' perpendicular to Oz is directed to the south, then OK is directedto the east. Angles z'Oz = B,^K = ,yVK = \p. GEOMETRY 33The direction cosines of the two systems of axes are given by the followingscheme : 34 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS ^"i, y>i, X3, Xi denote the distances of a point P from the four sides of a tet-rahedron (the tetrahedron of reference); k, mi, m; h, fth, fh'i h, ms, nz; andI4, nii, Hi the direction cosines of the normals to the planes Xi = o, X2 = o, x^ = o,Xa = o with respect to a rectangular system of coordinates x, y, z; and dx, d^, ds,di the distances of these 4 planes from the origin of coordinates : (i) Xi = hx + triiy + WiZ — diX2 = hx + nhy + fhz — diXi = ^3^ + mzy + naZ — dsXi = hx + m4y + n\Z — d\. s\, 52, 53, and 54 are the areas of the 4 faces of the tetrahedron of referenceand V its volume : 3F = .Ti5i + X^J^ + .T353 + .T454.By means of the first 3 equations of (i) x, y, z are determined : X = AiXi -\- B1X2 + CiXz + Di,y = A2X1 + B2X2 + C2.V3 + A,Z = A3X1 + B3X2 + C3.T3 + D3.The equation of any surface, F(x,y,z) = o,may be written in the homogeneous form : Fi \AyXi + B1X2 + C1X3 + -p (51.V1 + S2X2 + S3X3 + SiXi) , M2:>^1 + B2X2 + C2X3 + -|7 (SiXi + S2X2 + S3X3 + S4Xa) \,A 3X1 + B3X2 + C3X3 + -y (siXi + S2X2 + S3X3 + SiXi) PLANE GEOMETRY2.100 The equation of a line : Ax + By + C = o.2.101 If p is the perpendicular from the origin upon the Hne, and a and (3 theangles p makes with the .v- and j-axes : p = X cos a + y cos (3.2.102 If a' and (3' are the angles the line makes with the x- and >'-axes : p = y cos a' — X cos j3'.2.103 The equation of a line may be writteny = ax + b.a = tangent of angle the line makes with the .r-axis,b = intercept of the y-axis by the line. 35GEOMETRY2.104 The two lines: y = QiX^ bi,y = aox + bo,intersect at the point : bi — bi, a\bi — a^biX = y = •Gl — Oo Ol — O22.105 If (^ is the angle between the two lines 2.104 : , (?i - 02tan o = ± I + C1C22.106 Equations of two parallel lines : Ax -\- By + Ci -= o { y =- ax + b\,Ax + By -{- Co = o \y = ax + b^.2.107 Equations of two perpendicular Hnes :Ax + By -\- Ci = o ( y = ax + buorBx - Ay -{- C2 = o \ X ,\ y = - — + h.{ a2.108 Equation of line through Xi, yi and parallel to the line : Ax + By + C = o or y = ax -\- b,A {x - xi) -{- B{y - yi) = o or y - yi = a{x - X\), '2.109 Equation of line through Xi, yi and perpendicular to the lineAx -{- By -^ C = o or y = ax + b,B(x - xi) - A(y - yi) = o or y - >'i = •2.110 Equation of Hne through .Vi, 3-1 making an angle (/) with the line y = ax -{ b:a + tan o the singular point is an isolated point with no real branch of the curvethrough it.A = o the singular point is an osculating point, or a cusp. The curve has twobranches, with a common tangent, which meet at the singular point. ^, dF dF d'F d^'F d'F . . , . , ^ • . .u • 1If —' — ' -T-j' T-5' ^ , ^ simultaneously vanish at a point the singularpoint is one of higher order. PLANE CURVES, POLAR COORDINATES2.270 The equation of the curve is given in the form, r=f{d).In figure 2, OP = r, angle XOP = B, angle XTP = r, angle pPt = (j>.2.271 is measured in the counter-clockwise direction from the initial line,OXy and s, the arc, is so chosen as to increase with 6. The angle is measuredin the counter-clockwise direction from the positive radius vector to the positivetangent. Then, r = e + ^.2.272 tan — sin 4>),y = a(i — cos 0),where the x-axis is the base with the origin at the initial point of contact. 4> isthe angle turned through by the moving circle. (Fig. 6.) Fig.6A = vertex of cycloid.C = center of generating circle, drawn tangent at A.The tangent to the cycloid at P is parallel to the chord AQ.Arc AP = 2 X chord AQ.The radius of curvature at P is parallel to the chord QD and equal to 2 X chord QD.PQ = circular arc AQ.Length of cycloid: s = 8a; a = CA.Area of cycloid: 5 = 3x0-.2.451 A point on the radius, b>a, describes a prolate trochoid. A point,b,y = (a + d) (i - cos (j)),d = o Cycloid,d>o Prolate trochoid,d^ cos 0, /I. \ • A. • ^ ± ^ J.y = (o =h a) sin

')'2.466 Tractrix: 54 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS2.602 The perpendicular from the point Xi, >'i, Si upon the plane Ax + By +Cz + D = o is: - Axi + Bv, + Czi + Da =2.603 vOP + B' + C^is the angle between the two planes:Aix + Biy + Ciz + A = o,A^x + B^y + C2Z + A = o,AxAi + B,B2 + C1C2cos 62.604 Equation of the plane passing through the three points (xi, yi, Zi) {x2, yi, Z2)(xs, ys, S3):yi 2i I GEOMETRY 552.625 The perpendicular distance from the point Xo, y^, 22 to the Hne:X - .vi y - >'i s - ziis:d = { {xi-XiY + ()'2 - ji)- + (z2 - 2i)2}i - {/i(x2 - xi) + mi(y2 - yd + «i(z. - Zi)}2.626 The direction cosines of the hne passing through the two points Xi, yi, Zjand .V2, y2, 22 are: (V2 - -Vi), (>'2 - yi), (22 - Zi)|(.V2-Xi)^+(>'2->'l)^'+(22-Zl)2}^*2.627 The two Hnes:X = niiz + pi, X = nhz + p2,and . y = iiiz + qi, y = noZ + q^,intersect at a point if,{nil - nh) (qi - 92) - (wi - jpz) (pi - pi) = o.The coordinates of the point of intersection are: ^ ^ miP2-moPi ^ niq2-tu_qi ^ ^ po - pi ^ q^-qi ^nil - WZ2 ' Wi - ih ' mi - nh ni - n^The equation of the plane containing the two lines is then{ill - Ui) {x - miz - pi) = {nil - W.2) {y- niz - qi). SURFACES2.640 A single equation in x, y, z represents a surface:F{x, y, 2) = o.2.641 The direction cosines of the normal to the surface are:dF dF dF , dx dv dzI, m, n dFV 1 im-m-m2.642 The perpendicular from the origin upon the tangent plane at x, y, z is:p = Ix + my + nz.2.643 The two principal radii of curvature of the surface F {x, y, z) = o aregiven by the two roots of: 56 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS where: k dJFp dx' i GEOMETRY 572.649 The direction cosines to the normal to the surface in the form 2.648 aje: , d{u, v) ' d{u, v) ' diu, v)I, m, n — ^^ — ^^_/[diu, v)l ^U(«, v)) ^\d{u, v)) ]2.650 If the equation of the surface is:2 =fix, y),the equation of the tangent plane at Xi, yi, Zi is: 2.651 The direction cosines of the normal to the surface in the form 2.650 are: , -(i)-(g^ 'I, m, n = ^ — ^^^ -(IJ-(I)T12.652 The two principal radii of curvature of the surface in the form 2.650are given by the two roots of:(rt - 52)p2 - { (1+ q'y - 2pqs + (i + p-')i}Vi + f + q'^ p + (i + p2 + ^2)2 ^ ^^where df ^dj_ ^dj dH ay^ ax' ^ dy "^ ax2' ^~dxdy ^~a/'2.653 If Pi and p^ are the two principal radii of curvature of a surface, and pis the radius of curvature in a plane making an angle with the plane of pi,I cos^ - + z'')\(dl"\- (dm"Y /(9;?/'V]4 I 6o MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSdxds III. TRIGONOMETRYsm x I 1,13.00 tan x = > sec x = > esc x = -. > cot x = >cos X cos x sin x tan xsec- X = I + tan^a;, csc^x = i + cot-x, sin^x + coslx- = i,versin x = i - cos x, coversin x = i - sin x, haversin x = sin^(_,).y'i^2^. = y.3.01 sin X = - sin (- x) = V : > ^y cos^ - - cos" ->X2 tan - , X X tan X 2= 2 sm - cos - = — = = ,2 2 Vi + tan'^x , + tan^ 2^Vi + cof-^x ^Q^ ^ _ (,ot X tan - + cot x2 2 = cot -• (i - cos x) = tan -• (i + cos x), = sin y cos (x - y) + cos y sin (x - y),= cos V sin (x + y) - sin j cos (x + y), « «« / N 4 /l + cos 2X . , X 3.02 cos X = cos (- x) = y = 1-2 sin- -> , X . „ X X I= cos^ - - sm^ - = 2 cos^ I2 2 2 Vi + tan2 XI - tan- -2 I II + tan^ - I + tan x tan - tan x cot ^^— i2 2 2X Xcot - — tan -2 2 cot X sin 2Xcot ^ + tan :^ Vn- cotlx 2 sin x2 2 = cos y cos (x -f }») + sin y sin (x + y),= cos 3; cos (x — y)- sin 3^ sin (x - y),61 62 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSsin 2X I - COS 2X3.03 tan x = - tan (- x) I + cos 2X Sin 2X I^ — COS 2.V _ sin (x + y) + sin (.v — y)I + cos 2X COS (.v + }') + COS {x — y)'COS (x - y) - cos jx + y)sin (.V + >') - sin {x - y) cot a; - 2 cot 2X,tan tan X2 tan -2I - tan - I + tan - i - tan^ -2 2 2X XI - tan - I + tan -2 2 . I - e^'^ 3.04 The values of five trigonometric functions in terms of the sixth are givenin the following table. (For signs, see 3.05.) TRIGONOMETRY 63functions by the root of some quantity, the proper sign may be taken from thistable. 64 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS _ sin X _ I — COS X~ I + COS a; sin ic '±Vi + tan"^ .Y — Itan X 3.11 Functions of the Sum and Difference of Two Angles.3.111 sin {x ± y) = sin x cos y ± cos x sin y,= cos X cos y (tan x ± tan y),tan X ± tan 3^ . ^ ^ .= — sin (x T 3;),tan X T tan y ^ -^^' 3.114 - \ cos (x + y) + cos {x - y) \ (tan x ± tan y).3.112 cos (x ± y) = cos x cos y^ sin ic sin y,= cos X cos y (i ^ tan a; tan y),cot X =F tan y / -^ n= ^ cos ix T y) , cot X ± tan ycot y =F tan x . / _ s - = =^^ sin (x T y),cot y tan x =F i= cos X sin ;y (cot y T tan x).3.113 , , . tan x ± tan ytan (x ± 3') ^ cot (x ± y) I T tan X tan y TRIGONOMETRY 653.12 Sums and Differences of Trigonometric Functions.3.121 sin X ±sm y = 2 sin ^(.v i y) cos ^{x T y),= (cos .V- + cos y) tan ^{x ± y),= (cos y - cos .v) cot ^{x =F y),tan ^ (a- ± y) / . -^ . x= 7 w ^ X (sin X T sm y).tan ^ (x- T 3') '3.122 cos X + cos >> = 2 cos ^{x + y) cos K-^ - j)?sm X ± sm }>tan ^(x ± >>)'cot I (x + >')tan ^ (x - 3') (cos y — cos x).cos X - cos y = 2 sin | (>' + x) sin | (j - x)= —(sin X ± sin 3') tan |(x T j).sin (x ± y)tan X ± tan y cot X ± cot y = ± cos X • cos ysin (x ±y) , _^ , .= . ; ^'' (tan X T tan y),sm(x =F 3') = tan y tan (x ± }')(cot y =F tan x),I =F tan X tan y~ cot (x ± }') '= (i =F tan X tan y) tan (x ± 3;).sin (x ± y)sm X sm y sm X ± sm T 1 . .= tan l(x±y). - cotU--^-"^ y)-tan ^ (x + y) cos 66 MATHEMATICAL FORMULA AND ,ELLIPTIC FUNCTIONS3.1401. sin- X + sin- y = i - cos {x + y) cos (x - y).2. sin- X — sin- y = cos- y — cos- x= sin (x + )') sin {x - y).3. cos- X - sin- y = cos (.v + j) cos {x — y).4. sin- (.V + y) + sin- (^ - >») = i - cos ix cos a)*.5.' sin^ (x + )') - sin^ (x — y) = sin 2x sin 2j.6. cos- (x + y) + cos^ (x - y) = i + cos 2x cos 2y.7. cos^ (x -h y) — cos- (x -}') = — sin 2X sin ay. 3.1501. cos nx cos wx = | cos (« - m)x + ^ cos (w + m)x.2. sin »x sin mx = \ cos (« - w)x - ^ cos {n + w)x,3. cos nx sin mx = \ sin (« + m)x - | sin {n — m)x. 3.160I.2345678 gi+iy = gx ^(.Qg J _|- ^' sin }')•(cos X ± f sin x) " = cos nx ± i sin wx[De Moivre's Theorem],sin (x ± iy) = sin x cosh y ±i cos x sinh y.cos (x ± iy) = cos X cosh y =F i sin x sinh y.cos X = |(e'^ + e"'^).sin X = (c'^ - g *^).e^' = cos X + « sin x.g-ix ^ (,Qg ^ _ I sin -j;^ 3.170 Sines and Cosines of Multiple Angles.3.171 n an even integer:[ . (»- - 2-) . „ , (/?- - 2-) (;?- - 4-) . .sin nx = « cos x \ sin x ; sni-* x -\ ; sin^ x — .I 3! 5!w^ . , , n^{n" - 2-) . , n^i^n^ - 2^) (;?2 - ^^cos «x = I j sin- X H ; sin* x ^-^ -^ sin^ x + COS nx = cos .V \ I TRIGONOMETRY 673.172 n an odd integer: , . («2 - i^) . ^ , {n- - I-) («2 -32) -Isin nx = « { sin x j sin'^ x -\ j — sin^ x -....> • f {n- - I-) . „ (//- - I-) in- - 3") . , I- -^^ ; — - sin- .V + ^ —, — sin'* x -....) 'I 2! 4f J3.173 « an even integer: . ^^' / „ 1 • n 1 (^^ - 2) „ 3 . „ 3sin nx = (—1) cos x < 2"~^ sin"~^ x j— 2"""' sin"~"^ x(n-s)(n-4) ^„., ^j^„_, ^ _ (n-4)(n-5)(n-6) ^^_, ^.^2! 3! + . . . — r,n—S cin»—'= 1' -U —^ :^cos nx = (-1)2 { 2"~^ sin" x- ^, 2"'^ sin" ^ x + ""'" , ''" 2"-^ sin "~^ x^ ' [ i! 2!V ^^_v ^ 2 "-7 sin"-6 X + .3.174 n an odd integer : sin «x = (-1) 2 < 2"~* sm" X ; 2"-^ sin" ^ x -\ —^ 2"~^ sin"~* x[ i! 2!3! ^ N^^ / n I • n 1 « - 2 , . ,cos nx = (—1) 2 cos X < 2" ^ sin" ^ .v j— 2"~"^ sin""-^ a*(« - 3) (" - 4) ^„_5 ^i^„_5 _^ _ (» - 4) (n - 5) (^? - 6) ^„_72! 3! 3.175 n any integer: sin nx = sin x < 2"^^ cos"~^ x 7— 2"~^ cos"~^ x , (w - 3) (w - 4) „_5 „_5 (» - 4) (n - 5) (n - 6) , _|_ 2" ^ cos" ^ X — r^ 2" ' COS^+ . . .ft ft { 11 1. ICOS nx = 2"-l COS" X : 2"~^ COS""^ X + —^ 2""^ COS"~^ Xi! 2! —^^ ^--^ — 2 "-7 COS"-^ X + . .3! 68 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS3.176 sin 2x = 2 sin x cos .v.sin 3.V- = sin a- (3 - 4 sin- x)= sin .v(4 cos- x - i).sin 4x = sin x{8 cos^ x — 4 cos x).sin 5.T = sin x{s - 20 sin- .v* + 16 sin* x)= sin .v"(i6 cos'* x — 12 cos- x + i).sin 6.V = sin x(32 cos^ x - 32 cos^ x + 6 cos x).3.177 cos 2x = cos^ X - sin^ a;= 1 — 2 sin^ X= 2 cos- X - I.cos 3a; = cos .v(4 cos- x — 3)= cos x{i - 4 sin- x).cos 4X = 8 cos* x' — 8 cos- .t + i . cos s.^• = cos x(i6 cos* .v — 20 cos- .v + 5)= cos .v(i6 sin* x - 12 sin- a: + i)-cos 6x = 32 cos^ X - 48 cos* X + 18 cos- X - I. « ^„« 2 tan X3.178 tan 2x cot 2X = I — tan-^ Xcot- X — I2 cot X 3.180 Integral Powers of Sine and Cosine.3.181 11 an even integer : (—1)2 { , s nin — i) , .sm" X = —^^z-j [ cos «x - n cos (w - 2)x -{ j cos {n - 4)xn{n - i) in - 2) / ^x , . / ^^ 1; cos (« - 6)x + + (-1)- 2 if / . n{n — x) . ,cos" X = -^^j { cos nx + w cos \n - 2)x -\ ; — - cos {n - 4)x 'm2' ^ , nin - i) (« - 2) cos {n — 6)x + . . . -\- \^^ TRIGONOMETRY 693.182 n an odd integer:«— I (-1)"^ f . . / ,,«(«- l) . , Xsin" X = -—;^zr~ sni nx - n sm (« - 2)x H ^ sin (« - 4):^ cos" X = -;;zi COS ;7X + 11 cos (« - 2).v -\ —^ COS {n - ^)xn{n - 1) (n - 2) / .n , , nl . ] -\—:^ COS (« - 6).v + . . . . mm3.183 sin- X = h{i - cos 2.r).sin^ X = 1(3 sin x - sin 3.V).sin* X = I (cos 4.r - 4 cos 2X + 3) . sin^ X = iV(sin S-v - 5 sin 3.\- +10 sin x).X = — 3-V(cos 6x — 6 cos 4X+ 15 cos 2X — 10).sin3.184 COS" A- = ^(l + cos 2.v).cos^ a: = I (3 COS X + cos 3.V) . COS"* .r = 1(3 + 4 COS 2.V + cos 4.T).cos^ X = iV(io cos X + 5 COS 3.V + COS 5.v).COS^ X = 3^(10 + 15 COS 2X + 6 COS 4.V + COS 6x). INVERSE CIRCULAR FUNCTIONS3.20 The inverse circular and logarithmic functions are multiple valued; i.e., ifo> = sin ^{x7 ± >/(! - ^"'•^) (i - /)}cos ^{3'VI — XT T XV I — y'}.tan~i X ± tan~^ y = tan~^ _ ' •I =F xytan~^ X ± cot~^ y = tan"^ -^-——3' =F X ^-1 y^ X= cot^ x^y ± I HYPERBOLIC FUNCTIONS3.30 Formulas for the hyperbolic functions may be obtained from the corre-sponding formulas for the circular functions by replacing x by ix and using thefollowing relations:sin ix = \i{e'' - e'"") = i sinh x.cos ix = i (e ^ + g- ^) = cosh x.lie."'' — i)tan IX = —5 = I tanh x.e--^ + I cot tx = -I -7, = - I coth X.e-^ — I sec ix = — = sech x.e^ + e -^21 .CSC tx = = — t csch X.e^ — e "sin ^ ix = i sinh~^ x = i log (x + y/i + x-).cos-^ iX = - i cosh-^ X = i log (x + Vi + a--). tan~^ ix = « tanh"' x = i log V/ ^• cot~' ix = - i coth~i X = - i log y "^ ^ - 72 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS3.310 The values of five hyperbolic functions in terms of the sixth are given inthe following table : sinh X = a cosh tanh X = a coth X = a sech csch sinh cosh X tanh coth X sech csch \/a- — I Vi — ti"Vi + a- aVi + a~ Vi -a-Vo- - I \/d~ — IIa VI - a- vi — 0-Va- + I \/ar — I \/i — <.:'-Vi + a' Va'^ — I Vi TRIGONOMETRY 73 inh -X = Ys cosh .V , I t /cosh X + Icosh -X = V2 ' 21 cosh X — I sinh x _ /cosh .t — i2 " sinh .v cosh .v + i V cosh x + i sinh (.V ± y) = sinh x cosh y ± cosh x sinh y.cosh (.V- ± 3O = cosh X cosh }' ± sinh x sinh y.tanh X ± tanh ttanh (.V ± y) coth (x ± j) I ± tanh X tanh 3'coth X coth r =b icoth y ± coth a; sinh X + sinh v = 2 sinh l(x + y) cosh |(:r — 3')'sinh X — sinh y = 2 cosh ^(x + y) sinh |(x — 3')-cosh X + cosh y = 2 cosh ^(x + y) cosh |(x — 3').cosh X - cosh 3' = 2 sinh ^(x + 3') sinh ^{x - y).sinh (x + 3')t;. tanh x + tanh y = — —cosh X cosh y6. tanh x - tanh y = —; -r— ' cosh X cosh y ^, , .1 sinh (x + y)7. • coth X + coth V = ^—I . ' •sinh X smh yo ^-i. .1 sinh (x - v)8. coth X - coth 3' = ^-, r-r^smh X smh y 74 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 3.351. sinh (.V + 3') + sinh (x — y) = 2 sinh x cosh y.2. sinh (x + y) — sinh (x — y) =2 cosh x sinh y.3. cosh (a: + y) + cosh (x - y) = 2 cosh x cosh y.4. cosh (x- + y) - cosh (.v - y) = 2 sinh x sinh 3;. ^ , , . ^ sinh X ± sinh y5. tanh ^{x ±y) = — r-^ -cosh X + cosh y . ^1. 1 / \ sinh .T =F sinh y6. coth i {x ± 3') = — r^.cosh X — cosh ytanh .r + tanh y sinh (.r + r)- 8. tanh X - tanh y sinh (.r - 3').coth X + coth y _ sinh (.v + y)coth X — coth y sinh (x - 3;) 3.361. sinh {x + 3') + cosh {x + 3') = (cosh x + sinh x) (cosh 31 + sinh 3').2. sinh (:v' + 3^) sinh {x — y) = sinh- x - sinh- 3^= cosh^ X — cosh^ y.3. cosh {x + y) cosh (.r — y) = cosh^ x + sinh^ y= sinh- X + cosh^ y. . , , I + tanh \x4- sinh .r + cosh x = ;—;- •I — tanh f .V5. (sinh X + cosh x)" = cosh nx + sinh nx. 3.371. e^ = cosh X + sinh x.2. e~^ = cosh X - sinh x.3. sinh X = |(e^ — e~'').4. cosh X = |(e^ + e-^). TRIGONOMETRY 753.38I. sinh 2.V = 2 sinh x cosh x,2 tanh .VI - tanh" Xcosh 2X = cosh' X -|- sinh- x = 2 cosh-= 1 + 2 sinh- X,I + tanh- X tanh 2X = I — tanh^ X2 tanh a-I + tanh- X4. sinh 3.T = 3 sinh x + 4 sinh^ x.5. cosh 3X = 4 cosh^ x — 2 cosh :r. , T. tanh a: + tanh^ x6. tanh 3X = — :—r^ .^ 1 + 3 tanh^ X 3.40 Inverse Hyperbohc Functions.The hyperbohc functions being periodic, the inverse functions are multiplevalued (3.311). In the following formulas the periodic constants are omitted,the principal values only being given. I. sinh~^ X = log (x + V.v- + i) = cosh"^ v.v- + i.2. cosh~^ X = log {x + V.v- — i) = sinh ^ V .v- —tanh"' coth-^ 76 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 3.42I. cosh' i(.. + ^).sinh-.i(.v-i), = tanh~i '— = 2 tanh~i " ,x^ + I X + i'= log X.cosh"' CSC 2X = — sinh~^ cot 2X = — tanh~^ cos 2X,= log tan X.3. tanh~^ tan- (—h - ) = - log esc x.V4 2/ I ^X I4. tanh"^ tan^ - = - log sec x.2 2 3.43 The Gudermannian.If,1. cosh X = sec 6.2. sinh X = tan 9.3. e^ = sec ^ + tan = tan ( — + -fir d\4. X = log tan - + - I •\4 2/5. ^ = gd X. 3.44I.2.3- sinh .T = TRIGONOMETRY 776. tanh"^ tan x = | gd 2x.7. tan~^ tanh x = | gd~^ 2X. 3.50 SOLUTION OF OBLIQUE PLANE TRIANGLESa, b, c = Sides of triangle,a, 13, y = angles opposite to a, b, c, respectively,A = area of triangle,s = i(a + b-\-c).Given Sought Forinula a, b, c a, b, a b)(s-c)be/ sjs - a)2" V becos - a= V/ (s-h)is-c)tan I J is b e' + b^ ~ a"2beA = Vs{s - a){s - b){s - c). r. b sin aWhen a>b, Q<— and but one value results. When b>a 8 TRIGONOMETRY 79 3.52 Oblique-angled spherical triangles.a, b, c = sides of triangle. . a, /3, 7 = angles opposite to a, b, c, respectively.s = l{a + b + c),0- = H« + iS + T),e = a+jS+7 — i8o = spherical excess,^ = surface of triangle on sphere of radius r.Given 8o MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSGiven Sought Formulaj tan d = tan a cos y.\ sin (b - 6) = cot a tan y sin 6.1 , sin Ua+ y) ^, .tan - 6 = -:—J7 -. tan M« - c)2 sin |(a - 7)cos Ka + 7) ^ 1 . , .= Y^f \ tan \{fl + c).cos Ka - 7) a, b, y ctan = tan a cos 7tan 4) = tan 6 cos 7 c TRIGONOMETRY giGiven a, b, 7 a, b, c €, 7 Sought 82 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS "V^ . , , n cos (;/ + i).v-sin nxsin- kx = -O' TRIGONOMETRY 83n'^^ I X I .V16. 7,-7 tan—; = — cot 2 cot 2X.j6^ 2^ 2'' 2" 2"k=-o "V^ . P 27r Vn I nir . mr^18. 7 , sin = I + cos — — sin^^ n 2 \ 2k=i2. klT TTcot 2« ^^ I „ .V 2-""^- — I ., I X20. 7 .-7-, tan- -7 = T—T- + 4 cot- 2x ;- cot — •k—o 3.62 n— I 5'„ = ^ . CSC —Watson (Phil. Mag. 31, p. iii, 191 6) has obtained an asymptotic expansionfor this sum, and has given the following approximation:Sn = 2n {0.7329355992 logio(2«) - 0.1806453871}0.087266 0.01035 0.004 0.005n n^ 11^ ifValues of Sn are tabulated by integers from « = 2 to « = 30, and from n = 30to n= 100 at intervals of 5.The expansion of k= Iu 27r ^ ^ ^27rwhere 'perbolic Functions.The Smithsonian Mathematical Tables: Hyperbolic Functions, contain themost complete five-place tables of Hyperbolic Functions.Table I. The common logarithms (base 10) of sinh u, cosh u, tanh u, coth u:u = o.oooi to u = o.iooo interval o.oooi,ti = o.ooi to u = 3.000 interval o.ooi,u = 3.00 to u = 6.00 interval o.oi.Table II. sinh ti, cosh u, tanh u, coth u. Same ranges and intervals.Table III. sin u, cos u, logio sin u, logio cos u:u = 0.0001 to u = O.IOOO interval 0.0001,u = o.ioo to u = 1.600 interval o.ooi.Table IV. logioe" (7 places), e" and e~" (7 significant figures):u = O.OOI to M = 2.950 interval o.ooi,u = 3.00 to w = 6.00 interval o.oi,w = i.o to w = 100 interval i.o (9-10 figures).Table V. five-place table of natural logarithms, log u.u = 1.0 to u = 1000 interval i.o,u = 1000 to u = 10,000 varying intervals.Table VI. gd u (7 places) ; u expressed in radians, u = o.ooi to u = 3.000,interval o.ooi, and the corresponding angular measure, u = 3.00 to u = 6.00,interval o.oi.Table VII. gd~^u, to o'.oi, in terms of gd u in degrees and minutes from0° i' to 89° 59'.Table VIII. Table for conversion of radians into angular measure. TRIGONOMETRY 8g Kennelly: Tables of Complex Hyperbolic and Circular Functions,Cambridge, Harvard University Press, 1914.The complex argument, x + iq = pc'^. In the tables this is denoted pZ8.p = y/x^ -\- (f^ tan b = q/x.Tables I, II, III give the hyperbolic sine, cosine and tangent of {p/-b)expressed as r Z 7 : 6 = 45° to 5 = 90° interval 1°p = o.oi to p = 3.0 interval o.i. rr- 1 1 TT7- J ir • sinh ^ tanh , a ,5;Tables IV and V give —^— ,—^— expressed as rZ 7, t/ = pZo,p = 0.1 to p = 3.0 interval o.i,5 = 45° to 5 = 90° interval 1°.Table VI gives sinh (pZ 45°), cosh (pZ 45°), tanh (pZ 45°), coth (pZ 45°),sech (pZ 45°), csch (pZ 45°) expressed asrZ7:p = o to p = 6.0 interval 0.1,p = 6.05 to p = 20.50 interval 0.05.Tables VII, VIII and IX give sinh {x + iq), cosh {x + iq), tanh {x + iq),expressed as m + iv: X = o to X = 3.95 interval 0.05, ^ = o to 9 = 2.0 interval 0.05.Tables X, XI, XII give sinh {x + iq), cosh {x + iq), tanh {x + iq) expressedas rZ 7: X = o to a: = 3.95 interval 0.05,q = o io q = 2.0 interval 0.05.Table XIII gives sinh (4 + iq), cosh (4 + iq), tanh (4 + iq) expressed bothas u + iv and r Z 7 : q = o io q = 2.0 interval 0.05.Table XIV gives — and logio — •X = 4.00 to X = 10.00 interval o.oi.Table XV gives the real hyperbolic functions: sinh 6, cosh 6, tanh d, coth 6,sech 6, csch d. d = o to ^=2. 5 interval o.oi,d = 2.^ to 6 = 7.5 interval 0.1. 90 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSPernot and Woods: Logarithms of Hyperbolic Functions to 12 SignificantFigures. Berkeley, University of California Press, 1918.Table I. logio sinh x, with the first three differences, .V = .0000 to X = 2 018 nterval o.ooi.Table II. logio cosh x.X = 0.000 to X = 2.032 interval o.ooi.Table III. logio tanh x.X = 0.000 to X = 2.018 interval o.ooi.sinh XTable IV. logi Table V. logi xX = 0.00 to X = 0.506 interval o.ooi.tanh XXX = 0.000 to X = 0.506 interval o.ooi.Van Orstrand, Memoirs of the National Academy of Sciences, Vol. XIV,fifth memoir, Washington, 1921.Tables of — > e^, 6""=, e"'^, e~^^, e^T^, sin x, cos x, to 23-62 decimal places orsignificant figures. IV. VECTOR ANALYSIS4.000 A vector A has components along the three rectangular axes, x, y, z . ^ Xy Ay, A z. A = length of vector.A = V'AJTTT+'AJ. T-v- .• • i^AxAyAzDirection cosines oi A,— ,— , — . 4.001 Addition of vectors. A + B = C.C is a vector with components.C y = A y + By. 4.002 d = angle between A and B.C = VA-' + B' + 2AB cos d. . A,B, + AyBy+ AzBz^°^^ = AB4.003 If a, b, c are any three non-coplanar vectors of unit length, any vector^R, may be expressed: R = aa. + hh + re,where a, b, c are the lengths of the projections of R upon a, b, c respectively.4.004 Scalar product of two vectors:^AB = (AB) = ABare equivalent notations. AB = AB cos AB.4.005 Vector product of two vectors:TAB = A X B = [AB] = C.C is a vector whose length is C = ^5 sin AB.The direction of C is perpendicular to both A and B such that a right-handedrotation about C through the angle A B turns A into B.91 92 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS JJ = J" kk = k2 4.006 i, j, k are three unit vectors perpendicular to each other. If their direc-tions coincide with the axes x, y, s of a rectangular system of coordinates:4.007 4.008 4.0094.010 ij = ji = jk = kj = ki = ik = o. T'ij = - T'ji = k,Fjk = - I'kj = i,Fki = - rik = j.AB = BA = AB cos AB = A ^B, + A yBy + A ,B,FAB FBA i j kAx Ay ABx By B,(AyB,- A ,By)\ +{A.Bx- A x5,)j + (A ^By - A yBx))^. 4.10 If A, B, C, are any three vectors: "ATBC = BFCA = CP'AB= Volume of parallelepipedon having A, B, C as edgesA, VECTOR ANALYSIS 934.20 dAB = AdB+BdA.dVAB = VAdB + VdAB= VAdB - VBdA. 4.21 a . d d1— + ]^ +k-—dv azdx _, ,• * dA, dAy dA,VA = div A = ^~ +^ + -^- •dx ay azfVA VV = V- curl A = rot A 94 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS4.231. VAB = grad AB = (AV)B + (BV)A + I '.A curl B + I'.B curl A.2. V ^'AB = div FAB = B curl A - A curl B.3. FV FAB = (BV)A - (AV)B + A div B -B div A.4. div 4)k = 4) div A + AV0.5. cur! 4)k = V-\7(pA. + curl A = F-grad 0.A + curl A,6. VAr = 2(AV)A + 2 FA curl A.7. C(AV)B = A(CV)B + AFC curl B.8. BVA2 = 2A(BV)A. 4.24 R is a radius vector of length ;• and r a unit vector in the direction of R.R = rr, ,.2 = ^.2 + 3,2 + s2^v^ VECTOR ANALYSIS 95ds = an element of arc of a curve regarded as a vector whose direction isthat of the positive tangent to the curve. 4.31 Gauss's Theorem: fffdWAdV = ffAdS. 4.32 Green's Theorem: 4.33 Stokes's Theorem: // curl AJS = fMs. 4.40 A polar vector is one whose components, referred to a rectangular systemof axes, all change in sign when the three axes are reversed.4.401 An axial vector is one whose components are unchanged when the axesare reversed.4.402 The vector product of two polar or of two axial vectors is an axial vector.4.403 The vector product of a polar and an axial vector is a polar vector.4.404 The curl of a polar vector is an axial vector and the curl of an axial vectoris a polar vector.4.405 The scalar product of two polar or of two axial vectors is a true scalar,i.e., it keeps its sign if the axes to which the vectors are referred are reversed.4.406 The scalar product of an axial vector and a polar vector is a pseudo-scalar,i.e., it changes in sign when the axes of reference are reversed.4.407 The product or quotient of a polar vector and a true scalar is a polarvector; of an axial vector and a true scalar an axial vector; of a polar vectorand a pseudo-scalar an axial vector; of an axial vector and a pseudo-scalar apolar vector. 96 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS4.408 The gradient of a true scalar is a polar vector; the gradient of a pseudo-scalar is an axial vector.4.409 The divergence of a polar vector is a true scalar; of an axial vector apseudo-scalar. 4.6 Linear Vector Functions.4.610 A vector Q is a linear vector function of a vector R if its components,Qh Q2, (?3j along any three non-coplanar axes are linear functions of the com-ponents Ri, R2, R3 of R along the same axes.4.611 Linear Vector Operator. If co is the linear vector operator,Q = coR.This is equivalent to the three scalar equations,Qi = COnRi + Wi2^2 + OiisRs,Q2 = W21R1 + 6022^^2 + 0i2zRz,Qz = CCziRl + C032^2 + 0)3zR3.4.612 If a, b, C are the three non-coplanar unit axes, coii = S.aoJa, CO21 = ^.bcoa, 0)31 = 5.ccoa,CO12 = ^.acob, CO22 = 5.bcob, W32 = 5.ccob,CO13 = S.a.(Joc, CO23 = S.hicc 6033 = vS.ccoc.4.613 The conjugate linear vector operator co' is obtained from co by replacingcx)hk by cokh', h, k = I, 2, 3.4.614 In the symmetrical, or self-conjugate linear vector operator, denotedby CO, CO = |(co + co').Hence by 4.612 ^.acob = 5.bcoa, etc.4.615 The general linear vector function coR may always be resolved into thesum of a self-conjugate linear vector function of R and the vector product ofR by a vector c: coR = coR -f I .cR,where CO = 5(00 + co'),and C = KW32 - C023)i + K<^13 - C03l)j -f- K<^21 - C0i2)k,if i, j, k are three mutually perpendicular unit vectors.4.616 The general linear vector operator co may be determined by three non-coplanar vectors. A, B, C, where. VECTOR ANALYSIS 97A = acou + bcoi2 + CCO13,B = aco2i + bco22 + CCO23,C = acosi 4- bco32 + 00)33,and CO = a5.A + b5.B + cS.C.4.617 If CO is the general linear vector operator and co' its conjugate,coR = Rco',co'R = Rco 4.620 The symmetrical or self-conjugate linear vector operator has threemutually perpendicular axes. If these be taken along i, j, k,CO = i^.coii + j5'.C02J + IcS.cosk,where coi, CO2, CO3 are scalar quantities, the principal values of co.4.621 Referred to any system of three mutually perpendicular unit vectors,a, b, c, the self-conjugate operator, co, is determined by the three vectors (4.616):A = coa = acoii -f bcoi2 + ccon,B = cob = aco2i + boooo + CCO23,C = coc = aco3i + bco32 + CCO33,where CO/ifc = CO A/,,03 = a^.A + hS.B + cS.C.4.622 If n is one of the principal values, coi, CO2, CO3, these are given by the rootsof the cubic,n^ - n-(S.AsL + S.Bh + S.Cc) + n(S.3iVBC + ^.bTCA + S.cT'A^) - S.AVBC = o.4.623 In transforming from one to another system of rectangular axesthe following are invariant: ^.Aa + 6'.Bb + S.Cc = coi + CO2 + CO3.5aI'BC 4- 5.bFCA -1- 5.cFAB = CO2CO3 + CO3CO1 + CO1CO2.S.AT'BC = CO1CO2CO3.4.624COi -t- CO2 + CO3 = COii + CO22 + CO33,CO2CO3 + CO3CO1 + CO1CO2 = CO22CO33 + c033COn + CO11CO22 - co^ss - co^si -I- co^ia,CO1CO2CO3 = CO11CO22CO33 + 2CO23CO31CO12 - COiiCO-23 - CO22CO-3I - CO33CO-12.4.625 The principal axes of the self-conjugate operator, co, are those of thequadric : cou.T^ + C022y" + CO333" + 2C023ys + 2C03iG.r + 2o:i2xy = const.,where x, y, z are rectangular axes in the direction of a, b, c respectively. 98 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS4.626 Referred to its principal axes the equation of the quadric is, oJi.v- + W2}'" + W33- = const.4.627 Applying the self-conjugate operator, co, successively,coR = icOiT^i + jco2-/?2 + kusRs,cowR = co-R = ooi'Ri + ioi2^R2 + kw3-/?3,coco-R = oo^R = ioii^Ri + jc02^i?2 + kcoa'^/^s, co-iR = i— + j — + k — • Wi COo CO3 4.628 Applying a number of self-conjugate operators, a, jS, . . . ., all with thesame axes but with different principal values (aittotts), (jSi/^o/^s), ....aR = ia i?i + ja2^2 + kas^s,jSaR = ajSR = iai/5ii?i + ia^^'iRi + kasjSsi^a.4.629 ^.QcoR = 5.Rco(),= C0i()i7?i + CO2Q2R2 + 0)iQ:iR3. V. CURVILINEAR COORDINATES5.00 Given three surfaces. 3. u = Mx,y,z),V =f2{x,y,z),w = fsix, y, z).IX = 4>i{ll, V, w),y = 4>2{U, V, 2£'),3 = i d4>2 d4>2 d(f>3 d4>?,dw du dw du dw dud(f>i d(f)i d4>o d4>o d4>z 5(/)3du dv du dv du dv 5.01 The linear element of arc, ds, is given by:ds" = dx- ^- dy- + dz- = -j-^ + -^, + -r^ + 2gi dv dw + 2go dw du + 2^3 du dv.//i" //o" nz" 5.02 The surface elements, areas of parallelograms on the three surfaces, are:dvdw , .,, , .,dSu = -ri— V I - h-hfgi^,dw du , , .,. ^ ^^^v = ,. ,. VI - hz-hxgi-,dSu du dvh\h2 VI — h^lpfgi99 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 5.03 The volume of an elementary parallelepipedon is: 5.04 coi, 002, 003 are the angles between the normals to the surface /o, fz\ fz, f\\fi, /2 respectively: cos coi = h-Jtsgi,cos coo = hshgi,cos 0)3 = ^1^2g3. 5.05 Orthogonal Curvilinear Coordinates.gl = g2 = g3 = O, - „ du^ dv^ dw^ds- = -r^ + -7-; + -r-T'hx- h^ hi ,^ (fi; rfze) ,„ dxv du du dv^/?3 ^3/^1 «1«2du dv dwhihhi 5.06 h^, h'?, h^ are given by 5.00 (3) and also by: CURVILINEAR COORDINATES5.07 A vector, A, will have three components in the directions of the normalsto the orthogonal surfaces u, v, w:A = VA^ + .4^-' + A^,\ 5.08I. div A " ^' '' " \ du V//2//3 du) ^ dv [hs/h di'J + dw \hh dw) j / dv\hjcurlu, A = hJh d /.1„>du V//.1 5.09 The gradient of a scalar function, xj/, has three components in the directionsof the normals to the three orthogonal surfaces:OH dv dw5.20 Spherical Polar Coordinates. I. X < u = r,v = d, , W = (f>.r sin 6 cos ^)-^'} 5.22 Ellipsoidal Coordinates.u, V, w are the three roots of the equation: .r- V" z"+a>b>c, u>v>w.6 = u: Ellipsoid.6 = v: Hyperboloid of one sheet.6 = w: Hyperboloid of two sheets. CURVILINEAR COORDINATES ., (a- + ii) {a- + v) (a- + w) 103{a' I04 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS5.23 Conical Coordinates.The three orthogonal surfaces are: the spheres, the two cones:2. X- + y^ + z~ = u^^2 y2 _ ^2 .y2 _ ^2X2 1^2 32W2 2£;2 _ ^2 ^2 _ ^2^2>j,2>^2>^2^/ 2 _ Z^-fll- - ^2) (^2 - C") (t)2 _ ^2) (^2 _ ^2)M-(t;2 - k;2) //;',' = (^2 _ ^2) (f2 _ ^2)M''(li- — W^)5. //I = I, hy"6. div A = — — {u^Au) + —^—T^H — V\ ^ V ^ - ^ ^'w ou u{v- — w^) ov \+ r.. .. ^ Vr - w^ A^j^u{v^ — w'^) 7- V u^ du \ dujd \ Viv'~ - ^2) (c2 _ J,-;) aW2(-J^2 _ ^2-)V(6 -^ 5_ V(.-*-^j(.-.-^),^, 8. curl„ A = curl,, A U{V" — W") V(Z.2 - ^2) (^2 - ir) £ (V^i^^^^ .4,.) - V(62 - W) (r2 - 7£.2) A U/^iT^'' A^ I\/(/)2 - w') ic^ - w') dAu I d_U\/l curl,„ A = - ^-- ( uAu du dw u duV(Z'- -f') ic' - V') dAru^v^ - lip- ^^ 5.30 Elliptic Cylinder Coordinates.The three orthogonal surfaces are:I. The elliptic cylinders: c"u^ c-{u^ — l) CURVILINEAR COORDINATES I052. The hyperbolic cyHnders: '"cV c-(i - V-)3. The planes: s = w.2C is the distance between the foci of the confocal ellipses and hyperbolas:4. X = CUV.5. y = c-s/u^ — I Vi — v~.J} = j^ = ^^-^^ ^'3=1. 7- d-A = ^(^(£(^"^-^'^^") + l(-^^^^^^^'))/a- d-\ 32 curlu A = — -7— r— » cVii^ - V- dv dz , ^ a.u I a.4.curi„ A dz cy/ii^ - v^ ^u 5.31 Parabolic Cylinder Coordinates.The three orthogonal surfaces are the two parabolic cylinders:1. y- = 4CW.V + 4C-M-.2. y^ = — 4aa- + ^ch'^.And the planes:3. z = w.4. a; = c(t; — u).5. y = 2cVuv.I j< + i> I ?< + V ,6. 77, = , 7-:; = , h=i./ir u fh V I06 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS curl„ A = v/-^ dv u + V dz , . u dAu J u dA,curU A = — V —I T"u + v dz y u + V duV M\\ U + V / dv\\ u +u + V [ du ^ u I y V 5.40 Helical Coordinates. (Nicholson, Phil. Mag. 19, 77, 1910.)A cylinder of any cross-section is wound on a circular cyHnder in the form ofa helix of angle a. o = radius of circular cylinder on which the central line ofthe normal cross-sections of the helical cylinder lies. The z-axis is along theaxis of the cylinder of radius a.u = p and V = (f) are the polar coordinates in the plane of any normal sectionof the helical cylinder. 4> is measured from a line perpendicular to z and to thetangent to the cylinder.w = 6 = the twist in a plane perpendicular to z of the radius in that planemeasured from a line parallel to the x-axis: f X = (a + p cos (/)) cos ^ -t- p sin a sin 6 sin (f),I. \ y = (a + p cos 4") sin — p sin a cos 6 sin (f),[ z = a 6 tan a -h p cos o; sin 4>.hi = I, 7/2 = ->Phi d^ sec- a + 2ap cos 4> + p~{cos~ 4> + sin- a sin'-^) 5.50 Surfaces of Revolution,z-axis = axis of revolution.p, d = polar coordinates in any plane perpendicular to z-axis.1. ds" = dz" + dp- + p-dd^ _ du^ dip' dw^^ W^lh^^ hi'In any meridian plane, z, p, determine u, v, from:2. /(z -I- /p) = M -f- iv.3. ^^' = e.Then u, v, 6 will form a system of orthogonal curvilinear coordinates. CURVILINEAR COORDINATES I07 5.51 Spheroidal Coordinates (Prolate Spheroids):I. z + ip = c cosh (w + iv).j z = c cosh ti cos V,\ p = c sinh ti sin v.The three orthogonal surfaces are the ellipsoids and hyperboloids of revolution,and the planes, 6 : p.c^ cosh- u c- sinh^ u __f P'[ c- cos ^y c'^ sin- vWith cos z< = \, cos V = jj, : z = cX jJt,,p = fV(X-- I) (i-M'). 7 9 X- - I I - />t2c'iV- M') ^ t- (X^- M') " c-(X^ - i) (I - M'J 52 Spheroidal Coordinates (Oblate Spheroids);p + iz = c cosh(M + iv).z = c sinh u sin v.p = c cosh u cos z>.cosh w = X, cos V = fi. 6-(X-' - tx') - C2(X2 - iX') ' C2(X2 - l) (l - p:') 5.53 Parabolic Coordinates:I. z + ip = c(u + ivy.Z = C(i<2 _ 1)2)^ 3- u- = X, z;2 = ^^With curvilinear coordinates, X, ^i, d: I08 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONScy \ + iJL cy \ + iJL 2cV Xm 5.54 Toroidal Coordinates: , z + a + ipI. u + tv = log ^}'^ z- a + tp _ a sinh ucosh w — cos Va sin Vh = h cosh « — cos t^cosh u — cos z) , cosh u — cos 7;a a sinh mThe three orthogonal surfaces are:(a) Anchor rings, whose axial circles have radii,a coth u,and whose cross-sections are circles of radii,a csch w;(b) Spheres, whose centers are on the axis of revolution at distances,± a cot V,from the origin, whose radii are, a esc V,and which accordingly have a common circle,p = a, z = o;(c) Planes through the axis, w = 6 = consto VI. INFINITE SERIES6.00 An infinite series: CO2 W„ = Wi + M2 + «3 + . . . .is absolutely convergent if the series formed of the moduli of its terms:1mi| + |m2| + |W2|+....is convergent.A series which is convergent, but whose moduli do not form a convergentseries, is conditionally convergent.TESTS FOR CONVERGENCE6.011 Comparison test. The series 2w„ is absolutely convergent if ] w„ | isless than C | z;„ | where C is a number independent of ;?, and Vn is the,«th termof another series which is known to be absolutely convergent.6.012 Cauchy's test. If ILimit I I «the series Sm™ is absolutely convergent.6.013 D'Alembert's test. If for all values of n greater than some fixed value, r,the ratio —^ is less than p, where p is a positive number less than unityand independent of n, the series 2w„ is absolutely convergent.6.014 Cauchy's integral test. Let/(.v) be a steadily decreasing positive functionsuch that,Then the positive term series 2a„ is convergent if,fix)dx,is convergent.6.015 Raabe's test. The positive term series Xa„ is convergent if, ni -^— i]^l where /> I.It is divergent if, ^Un+l 7 ^I no MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS6.020 Alternating series. A series of real terms, alternately positive and nega-tive, is convergent if a„+i^a„ andlimit an = o.In such a series the sum of the first s terms differs from the sum of the series bya quantity less than the numerical value of the (s + i)st term.6.025 If —^ =1, the series Sm„ will be absolutely convergent ifthere is a positive number c, independent of n, such that,limit f I w„+i In —-^ - 1 } = - 1 - c.W—»a> u 6.030 The sum of an absolutely convergent series is not affected by changingthe order in which the terms occur.6.031 Two absolutely convergent series,S = til + U2 + U3+T = V1 + V2 + VS+may be multiplied together, and the sum of the products of their terms, writtenin any order, is ST, ST = u\Vi + ihvi + U1V2 +6.032 An absolutely convergent power series may be differentiated or iute-grated term by term and the resulting series will be absolutely convergent andequal to the differential or integral of the sum of the given series. 6.040 Uniform Convergence. An infinite series of functions of x,S{x) = Ui{x) + Uo{x) +Mz{x) +is uniformly convergent within a certain region of the variable x if a finite number,TV, can be found such that for all values of n ^N the absolute value of the remain-der, I Rn I after n terms is less than an assigned arbitrary small quantity e atall points within the given range.Example. The series. ^{i + x^yis absolutely convergent for all real values of x. Its sum is i + x- if x is not zero.If X is zero the sum is zero. The series is non-uniformly convergent in the neigh-borhood of a; = o. INFINITE SERIES III6.041 A uniformly convergent series is not necessarily absolutely convergent,nor is an absolutely convergent series necessarily uniformly convergent.6.042 A sufficient, though not necessary, test for uniform convergence is asfollows:If for all values of x within a certain region the moduli of the terms of theseries, 5 = ui(x) + ih{x) +are less than the corresponding terms of a convergent series of positive terms,r = il/i + Ah + Ms + . . . .where M n is independent of x, then the series S is uniformly convergent in thegiven region.6.043 A power series is uniformly convergent at all points within its circle ofconvergence.6.044 A uniformly convergent series,S = ui(x) + u^ix) +may be integrated term by term, and,f S dx = 2 fun{x) dx.n= I6.045 A uniformly convergent series,6" = Wi(.v) + uo{x) + . . . .may be differentiated term by term, and if the resulting series is uniformlyconvergent, oo 6.100 Taylor's theorem./U- + h) = f{x) + ^,/'(:v) + ~j"{x) + + ^"/(")(.r) + i?„.6.101 Lagrange's form for the remainder:\n -\- 1) . 6.102 Cauchy's form for the remainder:hn+X (j _ f)\,i 112 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS6.103fix) =m +f{ii)-'-^ +r(h)- ^^^ + . . . +p^\h) ^^^ + R,R^=fin+i){h + d {X - h)\ ^^' ~ ^'^"7 O<0{y).The expansion of f(y) in powers of x is:/w -/w + **(z)/'(2) + fi ^, [i<^»p/'(2)] SYMBOLIC REPRESENTATION OF INFINITE SERIES6.150 The infinite series:J{x) = I + oix + — a^x^ + — osx^ +.... + — akx" +2 . 3. R.may be written: fix) = .-,where a'' is interpreted as equivalent to a^.6.151 The infinite series, written without factorials,f(x) = I + aix + a^x^ + + akX^ + . . . .may be written: J{_x) = -^—,•' I - axwhere a'' is interpreted as equivalent to oa.6.152 Symbolic form of Taylor's theorem:f(x + h) = e''l/(.T).6.153 Taylor's theorem for functions of many variables:f(xi + hi, X2 + ho, . . . .) = e"'di:'^''"-dJl+ f{xi, X2, = f(x,X2,...)+hi^^ + h2^^+.... 2 ! dxi^ 2 ! " 6x1^x2 2 ! dx2^ INFINITE SERIES II3TRANSFORMATION OF INFINITE SERIESSeries which converge slowly may often be transformed to more rapidlyconverging series by the following methods.6.20 Euler's transformation formula:5 = Co + aix + QiX^ + = —^^0 + -^— ^j (-^ ) A^- ao,1 - X 1 - X J^ \ I - xjk=lwhere: Aoo = ai — ao,ArciQ = Aai — Aao = 02 - 201 + ao,A^co = A-ai - A-ao = as - 302 + 3^1 - ao, The second series may converge more rapidly than the first.Example i. ^ 114 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS6.22 Kummer's transformation.Ao, Ai, A2, .... is a sequence of positive numbers such that\ _ J A ^"'+tand Limit A^m,approaches a definite positive value. Usually this limit can be taken as unity.If not, it is only necessary to divide Am hy this limit:Limit .a= AmOm.Then: CO 00tn=n m= nE.xample i. OT= I . ,. m Limit ^Am = m. Km = , Am = I,m + I 7n—>co2jm^ ^'^2j(m+i )m'Applying the transformation to the series on the right:Am = —, Am = , a = o,2 m + 200 CO^;^= I +^+ ^ 2jm-(m+ i) (w + 2)'Applying the transformation n times:CO CO^ rn~ " '^ m-{m + i) (w + 2) , . . . {m + n)Example 2. ^ 2m - 1 . I ^ 2111Am = -, Am = ;— , a = o,2 2m + I5.I + V(-x)"-.^!2 AJ 4^2 - Im=i INFINITE SERIES II5Applying the transformation again, with:1 2m + I ^ 4W^ + IAm = , Am =^ , a = 0,2 2m — 1 4.m^ — IS = 1- 2Z^-)'Applying the transformation again, with: . I 2m + I . 4m^ + 3Am = , /^m = ^—0 -, a = o,2 2m — 3 4m- — 900in=iExample 3. 6- S(-)' (2W - l)^'m=i2m — 1 ^ 4m^ — 4W + iAm=—^ ^, Am = 7 TT , N' Oi = o,2{2ni - 3) {2m - 3) {2m + I)s-l + 4^(-^y {2tn — l) {2m + 3) (2W + i)^ 6.23 Leclert's modification of Rummer's transformation. With the samenotation as in 6.22 and, Limit A^m = 0), CO oc^ n= o f„=iExample i. 00 n=iflo = o, Am = I, CO = 2, a = O, Am = ,S=^ + -V (-1)— 1 ^ Il6 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSApplying Ihc Iransformation to the series on the right, with: , 2W + I . {2m + 1)2Co = o, Am = , A/n = 7 TT ;—:, w = 4, a = o,m - 1 (m - I) {m +2) 's = i2+ ^y (-1)- '24 2^^ m{m + 2) (2m + lY (2m + 3)2 6.26 Reversion of series. The power series:z = X - bix^ - b2X^ - bsx* —may be reversed, yielding:X = Z -\- CiZ^ + C2Z^ + CzZ"^ + .where : Ci = bi,C2 = 62 + 26i2,C3 = 63 + 5^1^2 + 5^1^C4 = 64 + 66i^>3 + 3^2^ + 2161262 + I46l^C5 = 65 + 7(6164 + 6263) + 28(61263 + 6i6o2) + 8461^62 + 4261^,C6 = 66 + 4(26165 + 26264 + 63^) + 12(361264 + 6616063 + 62^)+ 60(261263 + 3612622) + 33061^62 + i326i«,C7 = 67 + 9(6166 + 6265 + 6364) + 45(61^65 + 6163- + 62^63 + 2616264)+ 165(61^64 + 6i6o3 + 36126263) + 495(61^63 + 261W)+ 1 28761^^62 + 42961.^Van Orstrand (Phil. Mag. 19, 366, 19 10) gives the coefficients of the reversedseries up to C12. 6.30 Binomial series. , . n n(n - i) „ n(n — i) (n — 2) ,(i + xy = I+-X+ ^ , ^ x^ + — ^T^ ^ x^ + . . . .I 2! 3!{n-k)lk[ x''+ . . , . = I + (:)-('>-(:>'-(:>'-•• INFINITE SERIES II 76.31 Convergence of the binomial series.The series converges absolutely for \ x \ i.When .V- = i, the series converges for w> -i and diverges for n^—i. It is abso-lutely convergent only for n>o.When X = —I it is absolutely convergent for n>o, and divergent for ni put x = j in 6.30.\a\ a \a\ ^ b6.33 . . ,- n n(m - n) , n(m - n) (2m - n) ,I. (i + x)"' = I + -X ^—— x^ + -^ x^ -m 2\m^ 3!^"* , , .'' n(m - n) (2m - n) [{k- i)m - nl ,+ (-1) rr~k ^+ . . . .2. (l + X)-1 = I - X + X" - x^ + X* -3. (i + x)-2 = I - 2X + 3x2 _ ^^.3 + 5x4 - ... . 4. V7Tx^. + '-^-'-^.r'+'-^r^-l:l^.- + . . .2 2-4 2-4-6 2-4-6-8Vi + X 2 2-4 2-4-6 2-4-6-8*6. (l + X)i = I + - X 7 X2 + -^ X^ -^ x^ + . . . .3 3-6 3-6-9 3-6-9-I27. (i + x)-i = i-^x + ^x2-^^r3 + I4:7ll_o^._^__3 3-6 3-6-9 3-6-9-I28. (i+x)^ = i+Jx + ^x2-3:i:ix3+^^^x^- ^-^-"f ^ x^ + ....2 2-4 2-4-6 2-4-6-8 2-4-6-8-IO9. (i + x)-3 = I - ^ X +— x2 - ^^ x^ +2 2-4 2-4-610. (l + X)i = I + - .T - A ^2 +•-!- x3 _ _Z2_ ^4 + . . .4 32 128 20484 32 128 204812. (l -L t)= = I + - X X" -\ x^ -~ x"* + . . . .5 25 125 625 Il8 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS ^ ^ ^ 5 25 125 62514. {i+xY^^i + lx--^ x-2 + -^ x' - -^^ x' + . . .^ ^ ' ^ 5 72 1296 31104 ,5. (, +,.)-! = :_ i, + J:^._ ^,3+ Jli9.,._....•^ ^ ^ 6 72 1296 31104 6.350X X , 2X2 4.T'' S.T^ r 2/ -1I - X I + x I + x^ I + x^ 1 + x^ "- -^ 3- I - X I - X^ I - X* I - x^I 1,2,4 Cx20-X-I X+l'X^+l'X^+I6.351 w may be any real number. , ,\» «2 ^2 (^^2 _ 22) ,j2(^^2 _ 22) („2 _ ^2) 6.352 If a is a positive integer:II Ia a{a+ i) o(a + i) (o + 2)I , , (a - i) ! j ^ 'V •^"'6.353 If a and b are positive integers, and a-^-a+yfe- A=i «=i(Schwatt, Phil. Mag. 31, 75, 1916) INFINITE SERIES 119POLYNOMIAL SERIES6.360 6.361 6.362 6.363 ; : 5- 5- = —{co + CiX + C2X-2 +ao + aix + (Hx^ + asx? + .. . . ao . .), MATHEMATICAL F0RMULJ2 AND ELLIPTIC FUNCTIONS C3 = 03 + O1O2 + 7 (^l ,2^2 24 6.364 log (i + CiJC + a^-v^ + 03-^^ + . . .) = CiX -i- C2x" + Csx^ +Oi INFINITE SERIES 121 +2C2+3C3+(0 77—rn a^-Pa{'ar-az'' .... x''-^^' plciloil . . .The coeflficient of .v^' {k an integer) in the expansion of (i) is found by takingthe sum of all the terms (2) or (3) for the different combinations of p, Cifi,C3, , . . . whica satisfyCl + 2^2 + 3^3 + = ^,p + Ci + C2 + C3 + . . • . = w.cf. 6.361. In the following series the coefificients Bn are Bernoulli's numbers (6.902)and the coefficients £«, Euler's numbers (6.903).6.400 COX^ X^ ^ V^ f . ft:-"+^ r -> ^ -^3! 5! 7! ^^ (2«+l)!n=o002. cos .r = I - -. + -| - ri + • • • • = /j (-1) 7—Ti L^'=< 00 J.2! 4! 6! .^^ (2;?)'w= oI 2 17 62^. tan a; = :v + - x^ H x^ + — x'^ + -7^— x^ + . . . .^ 3 15 315 2835S^^^^^^^-- ^^<7]I X I „ 2 I4. cot X = x^ - X' - • x^^ 3 45 945 4725 X .^/ (2«)!H= I5. sec...i + Vv' + ^^*+^,*' + - ""St^,--'" ^•^<-1-^ 2! 4! 6^ A^ (2«)! |_ 4 Jn = o6. CSC X = - + ^ X + -^ x^ + -^, x^ + . . .^ 3!" 3-51 3-7 I 'ir^ 2('22"+i - i) - +X ^ ^, 5n+lX2«+' [X2< TT^].X .iJ^ (2w + 2) ! '- -6.41 1.-3 , 1-3 „. , I-3-5 „7I. sin~^ X = X 4-— x^ H ^ x^ + ^^-~- x^ + . . , .2-3 2-4-5 2-4-6-7 Z^22«(w!)2(2W+ l)^ -1 X^ (2w)! 2„,,COS 'I* = ^ ^—^^ ^ n+l MATHEMATICy\L FORMULA AND ELLIPTIC FUNCTIONS 2. tan~^ X = X— x^ + - .v" - - .r + .3 5 7 (Gregory's Series) x-^i = cot ^ X = / . (-i)" —2 LJ 2n + In=o COV y^ 2^»(.^!)^ / a-^ y ^,^^_I + .V-^ (2;j + i) ! Vi +WTT I I II4. tan 1^; = \ : ^ + "^2 X 3^^ 5ar 7.r 2 2^*^ '^"(2»+l)x2"+lTT I II 1-3 I , I-3-5 I^ 2 X 2-3.r 2-4-5X'' 2-4-6-7X'CO _ ^ _j _ TT "^ (2;?)! - - - CSC •^- - 2 -^ 22"(«!)" (2« + l) "" 6.42 , ,„ „ 2 x'^ 2-4 .r^ 2-4-6 .-v^1. (sin-i xY = x^^ + -^ - + -^ + . . .323-53 3-5-7 4 = y ?!:M^ ^v--LJ (2w + i)! (n + i)«= o2. (sin-1 x)^ = x^ + ^ 3^ (i + ^) x^ + fl 3^5'^ (^ + ^2 + ^5! V 3V 7! V 3- 5 .T^ +CO /> — I /*^Zi3. (tan- x) . = /..^ (-i)'-°- XXT^ 2n 2 ^*::t7^*o=I I , I916). .V'?) . log 2 + - .' -^ x' +j:^ .V - = log 2 - 7. (-1)" ,„ ,. '—Xk — ^^^ ' 124 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSI I-II I-I-3I4. log (l + Vl + X-) = log .T + :; + -00 5. log .T = (.T - i) - ^ (x - lY + -^(x- lY .2(-:)-(^ [o<.,; M= I ' * I X + I 3 Vi: + 1/ 5 Vj: + 1/ J^ 2» + I V» + 1/ L = 2 rv; + - x^ + - :\J^ +I -:» I 3 5 '^ 2« + I , x+ I j I I I I I 10. Vi + :k' log (^ + Vi + x') = --^ + - ^' -— ^' + 3.3.7 X' Vi + x^ 3 3-5 3-5-7 X2. |log(x + Vi + .-^)| =7-3^+^7 .x2<: ,^i^^ (2«+i)! L .1-2< I »i= o ;2 2 X* 2- \X^[ ~ 3 2 3-5 3S"^ .V.-. 2-"--(n- !)!(». -i)! x^» [0^1 '* 6 15. ( INFINITE SERIES 1 2513- 2 log (i + ^') = - ^i-V" - - sox^ + - S3X* - . . . U;2< I . where 5„ = - + - + - + . . .- (See 1.876).1 2 3 ni{log(:+.v)|^[^,.v3-i(i.. + ^4.'+i(^+^^+i4^'-••• H']-log (1 + x) / , \ /i , I \ ^'^—^-^ -—' = X - n(n + i) -< H —I +x)" ^ ' \n n+ 1/ 2!+ n{n + i) (« + 2) (i + -^ + —^)"\ - . . . . .v2< I •\n n+ I » + 2/3! L J o -2! 3! ^^n! 1 , (a- log a)- (.Tlogo)^2. a^ = I + X log a -h ^ ^—^ + -!^ 2_Z_ + . . .2! 3!3- e'' = g (i + .T + |j .v2 + ^ .r^ + ii .^4 + . . V x"^ ^x^ 8x^ , ^.r« s6-v^2! 4! 5! 6! 7!5. e^' £2 4^^ _ 3I£6 \2^ 4! 6! ^* • ' 7' , X^ ZX^ QX'^ '17X^2! 3! 4! 5! . -, X" 2X^ CV^7. e-"" '^ = I + X + - + —- + ^ + . . .2! 3! 4!8. e'«»"'- = i+x + -~^ + ^- . . .2 6 24 6.470 r:r2 __-— x2< 00 . INFINITE SERIES I 2700X- x^ x^ ^^ .T-" r ., , 1 .. cosh :.. I + -, + -, + -, + ... = 2/ 7t:;7! [•'-<°'J- 3. tanh X = .V - - .r^ + — .1^ -^ -v^ +"^ 3 ^5 315 =S(-)"-'^^f:o^---' [-tTIT 24. X COth .V = I + - :\2 - — :V^ 4- X^ -^ 3 45 945 _ -^ 2 24 720 jL^ {2n)\ L 4j«= I6. X csch :*; = I - 7 :v-2 + 4- -^^ - ^^^ •^•' + • •6 360 15120 I +S(-)"^^iSf^--^" ^=<^-6.475 22 2^ 2^1. cosh a; cos ,T = I - — :y^ + — x^ - —, .t^^ + . • .2" 2^ 2®2. sinh X sin :» = — x^ - — .v^ + -^ x^'' -2! 6! 10!6.4761. gXcosO (.Qg (^. gin d) = Tv j .V-< I • 2. g^^"'^ sin (.r sm 0) = /^ j .v-< i ax / • /3N "^ x^" CO?, 2nd r "^ 13. cosh (x cos ^) • cos {x sin C;) = 7.—7-—-j — .r- < i • •ur m ^ • m X^ a:^"+^cos(2w + i)^ [v'-^tI4. sinh (x cos t^) • cos (x sin C^) = 7. j^ x , .v < i«= om . / • m X^x-"+'sin (2;/ + i)^ r -/ 15. cosh (x cos d) sin (x sin B) = 2j {2,1 + i) 1 L'^" J ' M=--0 • , / m • / • m X^:^-" sin 2nB \ » ^ ~\6. sinh (x cos 0) • sin (x sin H) = 7^—7—yj x-< in=i 128 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 6.480I. sinh~^ X = X — — x^ H — x^2-3 2-4-5 -Z^-')%^-.(«yt+.) ^"" h\ •1-1 1 II I • ^ I2. smh ^ X = lo" 2x -\ - —- —T + . . .2 2X'' 2-4 4.V^(2W) Ilog^+S^-'-^^ffe,^"^" u^>.w=ou-i 1 II i'3 I3. cosh 1 ^ = lOff 2:J(; r ^ — r 2 2.T^ 2-4 4X*"^ (2;z)t rJmJ 2^"{n^)^2n |_n= oIII "^^ ^.2n+l r3 5 7 AJ 211+ 1 \_ . , , I I II i-^ I5. sinh-i - = -^ +^^X x 2 2,x 2-4 sxcsch-ix = 'y (-1)" .„, ,(f'^' ^ . x-^"-^ L2>i.^ 2'"(w!)- (2W +1) L»= oI x^ I • 3 X*22 2-4 4 CO= sech-1 :*: = log - - ^ ,„. .x, x^" x2< in= o1 x^ I • 3 X*2 2 2-4 4 = csch-1 X = log - + ^ . (-1)" .,, . ,J X^" x2< I «=o coth-ix= "^-^—f^ rx2>i^mJ 2% + 1 L 6. cosh~^ - = log —^ .X X 4 . , , I , 2 I 2 1-7. smh-i - = log-H — ^ 1-.-. . .X X2 4 tanh-.i.'+-L + ^, + . INFINITE SERIES 1 296.490 1. _JL_ = Ve-(-+i).2 Sinn X A0^02. —L^ = 'V(-i)",-(-+^).2 cosh X ^^CO3. ^(tanhx-i)=2^-^)"^"'" 004. - - log tanh - = 'y^j—^—e-' ^"'+'^^2 2 ^^ 2« + I6.491 « = I 1, « = I JBy means of this formula a slowly converging series may be transformedinto a rapidly converging series.6.495 I. tan -v = 2x\ /TT ^- +^-^ +^^^ + ... I 2X2. cot X = —^ CDX^ 8x~Zj {2n - i)V - 4x''^« = I CO2x 2.r I "^^a2 ~ (27r)2 - ^2 - (37r)2 - x2 ~ ' * ' ' = ^ ~ Z^;^52 M = I3. sec X = -—!-, 7 r^i + {211 - i)-7r"^ — \x^n = I _ _ I 2X 2.V 2X4. esc X — H 5 ^ p TT :; + 7 ri; ^ — . . . •X TT^ - X^ (27r)2 - X- (37r)' - X^X jLJ^ n-TT- - X-n = IBy replacing x by ix the corresponding series for the hyperbolic functionsmay be written. 130 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSINFINITE PRODUCTS6.50 I. sin :^ = :^n(^-;^)- » = 1 3. cos .= n ('- (.,, +1)'T')'n = o004. cosh ^= n('+(»fi)v)- 6.51 ^^ =nsin X TT ^cos —2" ' I - X JJ(i+:*;2«). [x^Ki-], 6.63 1. cosh X - cos _y = 2 ( i + —J sitf ^ TT ( i + 7 —r-, ) ( i + 7 r,) • M = I 2. COS X - cos y = 2 I -] sin^ - I I i - 7 ; — tt> i - 7 ^9 ' 6.55 The convergent infinite series: 00I +Ui+th+ ' • .= 1+^ Wn. INFINITE SERIES • I3Imay be transformed into the infinite product(i + Vi) (i + V2) (i + V3). ....00 =n ^^ + ^")'where I + «1 + «2 + . . . . + Un-i 6.600 The Gamma Function:r(.)=-;f[('^")Z-I-J. 2 '»=i i + -nz may have any real or complex value, except o, — i, —2, —3,6.601 n=i6.602 Limit f I I 1,17= I+- + - + . ... + -- log m 6.603 r(2 + i) = 2r(2),sm TTZ6.604 For z real and positive = x:T{x) = Jj e-' t'-' at,log r(i + ^) = (x + ^j log X - a; + ^ log 27r + / | -^^ -J + l6.605 If 2 = n, a positive integer:T(n) = (n-i)l,v/^ ,A I-3-5 . . (2W- i ) ._ e 132 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS6.606 The Beta Function. If x and y are real and positive:p/ ^ r>. X T(x) r(v)B(x,y) = B(y,x)= Y{x + y) 'B(x,y) =£ t'-^ {1 - t)y-' dt,B{x^i,y) = -^B{x,y),X -f- y 6.610 For X real and positive:T'{x)6.611 6.612 n= o ^P{x+l)=l + ^Pix),l/'(l - x) = 4^(x) + T cot TX. 6.613 6.620 6.621 6.622 INFINITE SERIES 133 /3(i) = log 2,Ki)-:- 6.630 Gauss's 11 Function: '•n(*,^)-*'lL + „-n=i2. n {k,z + i) = n (k,z)- 3. n(.) = l:^^^n(^,s). 134 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSDarling (Quarterly Journal, 49, p. 36, 1920) has obtained an approximationto this integral: -—^tan-1 e^^d + xh-^^^y2 VttFresnel's Integrals:r ^ (-1)^6.701 X cos (..');'2fc3-5. . 7(4^ + 1)k:=o •^-^ ^2k+ly4k+ 33-5 (4^ + 3)* = o 16.703 r ^,rf/ = y] (-1)"—^;« = o6.704 _^,r'i!:!iL^-rf((^ - i)!Jo I - x/*"00 oc^{a + nb) (a + nb + i) (a + nb + 2) . . . (a + «6 + ^ - i)lb>o, .-v^^i].(Special cases, 6.445 and 6.923).CD 00ZJ ^ nl{n + y) JLJ y{y + i) . . . (j + n)n = a w = o6.706 If the sum of the series, 00j{x) = ^cnx- Co<.r6] INFINITE SERIES I3500 00pea p2ir -ysry6.707 I f(x)2j - sin n.vdx = - j (tt - /)V/(/ + 2mr)-dLM = I n = oExample i. /(x) = e"^^ lk>6].00I. r+2^^ 7:7-—5 = TT g*7r _^ g- Replacing j^ by -, and subtracting, I .'^ ._ v„ I 27r yfe + 2^^ (-ij ^2 _^ ^^2 - g^TT _ ^-kn-n = IExample 2. With f(x) = e"^^cos^tx and e"^"' sin [jlx. -^+yl ^ + ^ U-X'^ + M" .^^ X- + (w — ^t)^ X^ + (w + ^t)'^ cosh 2X7r — cos 2/i7rTTsinh 2\t °° r 1 •fJL "^ J n - 11 n + IJL [ ^ TTsm 2/x7rX^ + jJL" ^J I X" + (w — /x)^ X^ + {ji + ^t)- I cosh 2X7r — cos 2/i7r6.709 If the sum of the series, 00 n = ois known, then /»ao/ X / X / N / e-Hy-HU)dtan + aiy + as}'!)' + i) + Osyiy + 1) {y + 2) + = Jo •6.710 The complete elliptic integral of the first kind:J oV{i- x') (i - kv) Jo Vi - k" sin2 e n= IIf ^' = 1-Vi-k^7r(i + ^') f , /i 136 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS6.711 The complete elliptic integral of the second kind:dd.^ = I Vi- k^ sin' d 2 [ \2/ I \2-4/ 3CO -2T f "'S^ /i-3-5 • • • (-'I — i)V k^"2 [ A^ \ 2 •4-6 .... 2« / 211 - I I + VI - *- Fourier's series6.800 If f(x) is uniformly convergent in the interval: -c 4 4 a/wC'iy^ . . ^j_j I . 2(2;? - i)7r/w = ^2; (-1)" sm X.TT^ ^^ (2W - l)2 Cn = I6.804 /(x) = mx, - - < X < +-•' 2 2 = tn{x-c), +- < X <^,2 2«^-)=?s ( — i)" ^ . 2;zx.i;smn cn -^ I6.805 fix) = -a, -sb ^x ^ - sb, = '^(x + 2b), -^b^x^-b, = a, - b ^ X ^ + b, = -^(x-2b), b^x^ sb, = -a, 3& ^ X ^ 56. f{X) = — { cos -^^ cos -^-^^ ;:; COS ~- + — COSSTTX I 77rx I 77ra;- „ ^„. - r, cos^ - cos ^-r-46 3- 46 5- 46 7- 46 138 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS6.806 /(x)=^.T + 6, -l^x^o, = --x+b, o^x^l.8b X^ I / . TTXf{x) =—./. 7 \—yy cos (2« + i) — 7 ^ if-^ {an +1)- 2I6.807 Kx)=-^x, o^x^b, ,. , 2aP "^^ I . mrb . mrx/w = ~:^rri—r\ / j ~~^ sm —— sm —t--^ 7r-6(/ - h) L^ n- I I 6.810 X = 2^^ sin «:j; - tt< v< tt • 6.811 cos flx = — sin ott < h a^ ,—^ r cos m.t > -7r 1 / • > \ "NT^ C0S2(«+i)a:6.823 cos 2x - — - .V sin 2x + sin- a* log (Asin- x) = 7 . j^ ~\2 / & ^t ^ ^ «(w+l)[o ^ .1- ^ tt].6.824 sin 2x - (tt — 2x)sin- x — sin x cos x log (4sin- .v)00 _ '^ sin 2 (w + i)x r X / 1^^ n{n + i) L Jn= I00 - --_ I TT . '%^^ cos 2nx r ttI6.825 - - - sin x = > . 7 r-, r o ^ t ^ - •24 AJ {2n - l) (2« +1) L ^ ^ 2 I e Qon ^ sin .r V^ „ •6.830 = 7 , r"^ sin nxI — 2r cos X + r~ ^ 0^06.831 tan~^ '-— = / . - >'" sin «.r r "" cos wx r-< I •I — 2r cos X + r- •*" w L J6.834 "og -V^ 14© MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS006.835 - tan-' ^I^^li^ = V (_i)"-i -^^^ cos (2« - i);t; \r''3:) » ^4 = — = 1.0823232337 59 = = 1.0020083928,90 29749.35 ... c, TT^ , 5io = I.OOO9945751,^' = l^^J^S = ^•°36927755i ^,, = 1.0004941886.6.901 III3" 5" 7"TTWi = -J4 • • ^ *^ '^' ' (2>^+l)»'M2 = 0.9159656 . . .Ui = 0.98894455 ....Me = 0.99868522 ....A table of «„ from » = i to w = 38 to 18 decimal places is given by Glaisher,Messenger of Mathematics, 42, p. 49, 1913.6.902 Bernoulli's Numbers. CO22n-1^2n i i j j "^^ I(2W)! I^" 2-" 32" 42" ^i^ F»k = I ^ (22" - l)7r2" ^ 1,1,1„ _ I I I I "^^ I2(2W)! ^" = 7^' + J^ + ^ + ^ + • • • • ^ Zj i2k+ l)-« 'k = o(2«)! " l2" 22» + 32» ^ln + - • • ' - j^^ ^) k'nk = IB. = i, i?3 = i, .6 42Bi = — > Bi = — > 30 30 INFINITE SERIES 141R 3617^^- X>8 = '66 510691 ^ ^ 43867^2730 ' 798 '7 1746117» X>io =6 330 R 5 6.903 Euler's NumbersIn+l22n+2(2^)! " 32«+l 52n+ l 72^+1 ^^ ^ ' {2k - 1)'''^+'k = IEl =1, £4 = 1385,£2 =5, £5 = 50521,£3 = 61, Ee = 2702765.6.904^ 2«(2W - l) p , 2«(2W - l) (2n - 2) (211 3)£« - j En-1 + j £n-2 - . . . .•2! 4! - + (-1)" = O.6.905 'IlifljlA B^ = (2« - !)£„_, - ^^" - '^ ^''' 7 '^ ^''' - 3^ £„_,2)1 3 ! (2;? - l) (2W - 2) (2;? - 3) (2;? - 4) {2)1 - 5) , / X 1+ ^I ^"-3 - + (-1)" . 6.910 142 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS6.912 I. log 2 =/. -^—^ 2. INFINITE SERIES I4300 n=i CDK= I 2 A^ " (2« - l) {2)1 + 2) TT 2 ^/ (2;? - l) (2« + 2)00 2 TT' i^^ 4» + I(2W — l) {211 + 2)6.916If m is an integer, and n = mis excluded from the summation: 2. ^, n=i = y-iAJ wr — n^W=I00S ~ ^ - (weven).6.917 "^n - I I2. -2 ^^ 4.n-4^^'-i*00 6.918 144 VII. SPECIAL APPLICATIONS OFANALYSIS.7.10 Indeterminate Forms.7.101 -. If \',\ assumes the indeterminate value - for x = a, the true valueo l{x) oof the quotient may be found by replacing /(.r) and F{;x) by their developmentsin series, if valid for x = a.Example : r sin- X ~\|_i - cos a'Jx=o' _ 3! / ^ V 3!I — COS X Xr X* I X^2! 4! 2! 4!Therefore, r sin- X 1[l - COS .vJx=o7.102 L'Hospital's Rule. If /(o + h) and F{a + //) can be developed by Taylor'sfix)Theorem (6.100) then the true value of '-rrr^ for x = a is,/' \x) F'ia)provided that this has a definite value (o, finite, or infinite). If the ratio of thefirst derivatives is still indeterminate, the true value may be found by takingthat of the ratio of the first one of the higher derivatives that is definite.f(x)7.103 The true value of 777^ for a- = a is the limit, for h = o, of pi F''^)(a)where /^p) (a) and F ^'^ (a) are the first of the higher derivatives of /(.v) and F(x)that do not vanish for x = a. The true value of ;,/ , for x =0 is o if /?>o, 0° ifF(x)f(p) (a)poo /(.v) ^ J\ J \ JIf ,, . = + I, the expression has the indeterminate form 0° X o whichmay be treated by 7.120. 7.140 I 00 , 0°, 00 ". If { F(x) I '•'''^^ is indeterminate in any of these forms for x = a,its true value may be found by finding the true value of the logarithm of thegiven expression.Example; n. ' y, logy = -tanx-log X, 148 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS x = SPECIAL APPLICATIONS OF ANALYSIS I497.17 Special Indeterminate Forms and Limiting Values. In the following thenotation [/(.v)]„ means the limit approached by/(x) as x approaches a as a limit.7.171 1.(1 + -) =6'' (c a constant). 2.. [V.v + c - V.v]^ = o. 3. [yx{x + c) - a:]^ = - 4. [y{x + fi) {x + fo) - x]^ = h{ci + C2). 5. i / (X + Ci) (.V + 6-2) ... . (.T + C„) - X' = - (fi + C2 + . . . Cn). 7. [log(o + ...<;')-log(i+i)]^=l. 10. =00 (t7>l). 11. ~ =0 (.r a positive integer). -pi-[log a:l —i = °-14. (a + k")^" = c (c>i). La + ^x" ^ ^ ^ 'Jco /3 17. ^ (a + ^.r-) « + ^'"^*J^= J (w>o). ISO7.172 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS[X sin - [ COS - ) = o.Xj J CO cot- 8. sin- • log {a + be'^)|_ X Joo 10. (i + a tan-j = e"' 11. cos - + a sin - =L\ X X) Joo 7.173 SPECIAL APPLICATIONS OF ANALYSIS 1517.175 1. [x^] = -^- 5- [cos- ^^ tan ^] =002. [(tT - 2x)tan X]7r = 2. 6. [(a + be tanx)7r-2x-]^^ ^ g2_ - 2 4. [(.^ - .Otan ^"1 = ^ e^. 8. [(tan a:)^^'" ^^> = i.L ^ 2C Jc TT 46 7.18 Limiting Values of Sums.Limit/i^- + 2* + 3* + . . . . + «n I .. , ^00 if ^< — I. 2. "^""'^(^ + 1— + 1—. + . . . . + -1 -)Limit / I I I«—> 00 \na na -\- h na + 2b nalog (a + b) - log a (a, b>6).Limit/ n - i^ n - 2^ w — 3-w^ooVi.2-(w+ i) 2-3-(/i+2) 3-4-(w + 3)+ .-(. + i)-(.+ ;OJ^^-^°^^-Limit r/ , ViV / ^ ^ \/;y / v^\24. U + 6— + a- + &— + a3 + 6—^ +n-^ c» [_\ n I \ n I \ n )i n , .^«V1 «' , ^'+ U» + & = 5 + - ,\ n I \ I — a- 2if a is a positive proper fraction. 5. ^'"•"[v/^ +v/^ + v/«^ + + \/^l = ".if 6>o and a is a positive proper fraction. ^ Limitf, / r , n T , n r , / ri/J—>oo|_V i-« V 2-« V 3-» V n-nJ+ 2\/b,I - Vaif 6>o and a is a positive proper fraction.Limit r II, I , "II + - + - + . . . .+--log« =7 = 0-5772157 • •n-^ co[_ 23 n J J//J/ (6.602). 152 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 7.19 Limiting Values of Products. , Limit r/ ^ A / +^) (. + _f_) ....(, + ^_)1 . ,,n-^col\ nj\ n+i/\ « + 2/ V 211-1/] 'if c>o. w^oo|_\ naj\ na + b)\ na + ah) ' ' ' -y na + (n - i)b) j if a, b, c are all positive.Limitnw(w + i) (w + 2) (w + w - i)}""] I + -a2 if m>o. -::rl(-i)(-^)(-|) (^H-f)]=.. 7.20 Maxima and Minima.7.201 Functions of One Variable, y = f(x) is a maximum or minimum for thevalues of x satisfying the equation, f'{x) = -4^^ = o,oxprovided that f'(x) is continuous for these values of x.7.202 If, for X = a, f(a) = o,y =/(a) is a maximum if J"{a)o.Example: v = , /3>o,/'(x) -' + "(x2 + ax + )S)2'/'(a:) = o when x = ±\//3,2x^ - 6l3x - 2 a/?/'W = (x^ + ax + (3yFor X = -\-VB, fix) = —n = Maximum, SPECIAL APPLICATIONS OF ANALYSIS 153For X v/3, r(x) = ±;| Minimum, Iymin = 7^'a - 2V/37.203 If for X = a, /'(a) = o and f"(a) = o, in order to determine whethery = f(a) is a maximum or minimum it is necessary to form the higher differentialcoefficients, until one of even order is found which does not vanish for x = a.y = f{a) is a maximum or minimum according as the first of the differentialcoefficients, J" (a), /'(a), /^(a), of even order which does not vanishis negative or positive. 7.210 Functions of Two Variables. F(x, y) is a maximum or minimum for thepair of values of x and y that satisfy the equations, and for which dFdx _^F_dx dy dF dx- ay-d"F d~FIf both --7, and -r-r, are negative for this pair of values of x and y, F(x, y) isa maximum. If they are both positive F{x, y) is a minimum. 7.220 Functions of n Variables. For the maximum or minimum of a functionof n variables, F(.Vi, .r2 , Xn), it is necessary that the first derivatives,dF , -r-^ all vanish; and that the lowest order of the higher deriv-dF dFdxi dX'i 'atives which do not all vanish be an even number. If this number be 2 thenecessary condition for a minimum is that all of the determinants,D,= /u /. f2k k= I, where fki ft fij = • fkkd-Fdxi dx/ 154 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSshall be positive. For a maximum the determinants must be alternately negativeand positive, beginning with Di = ^-^ negative. 7.230 Maxima and Minima with Conditions. If F{x\, Xz, , Xn) is tobe made a maximum or minimum subject to the conditions, ' (jiliXi, CVo, , Xn) = O(t>i{Xi,X2, ,Xn) = O ^(()k{Xi,X2, ,Xn)where k{l, ni, n) = I- + ni- + w- -1=0.This is the same as finding the minima and maxima ofF{1, m, n) = UnP + 022"^^ + 033"^ + 2ai2lm + 2aizmn + laizln.Equation (2) gives: (flu + \)l + 012m + ai3« = o,012/ + (ct22 + \)m -\- a'2sn = o,Oi3^ + 023^ + (ass + X)n = o.Multiplying these 3 equations by I, m, n respectively and adding, y2 SPECIAL APPLICATIONS OF ANALYSISThen by (i. 1.363) the 3 values of r are given by the 3 roots ofIan - —-, an an an ai3 O22— 5 «23 C23 Qsz - 155 7.30 Derivatives. 7.31 First Derivatives. 1. -.— = wx" \dx''da -^ ,2. — = a^ log a. .de^3- ^ = ^^• Jxdx dx' / , NJ loga :r _ I _ loga edx X log a X . d log X I6. —J— = -•ax X2x'°° ^"^ log X.(log x) ^"M I + log -i;- log log x} . x\ ^dx \ ed sin X log X. 13- dxd cos Xdxd tan Xdxd cot Xdxd sec X „14. —;— = sec^ x-sin x.dx d esc X15- 156 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSd tanh"^ x d coth~i x i21. SPECIAL APPLICATIONS OF ANALYSIS 1577.351 Leibnitz's Theorem. If u and v are functions of x,d"(uv) d"v n dii d"~'^v n(n — i) d^u d"~^vdx"- dx" ildxdx"~^ 2! dx^ dx"~-n(n — i) (n — 2) d^u d"~^v d"u3 ! dx^ dx'^"^ dx^7.352 Symbolically, where w° = u, v^ = V.7.353 ___,„^, + ^j„.7.354 If (t~.) is a polynomial in — , ^(J-yu = e-4>[a + [y7.355 Euler's Theorem. If m is a homogeneous function of the »th degree of rvariables, Xi, xo, . . . x,-,/ d d d \'«\ dXi dX2 dXrJwhere m may be any integer, including o. 7.36 Derivatives of Functions of Functions.7.361 If f(x) = F(y), and y = (/)(x), where27.362 dx" \x/ X-" \xj .v-^"~^ II. (_i).i^ F (^) = ^, F'") (^) + ^^^ ^. F^"-) [lin - i) {n - 2) n{n - i) ^(,^_2) /^i\ , _^2»-2 2! 2!+ (w - i) (w - 2) (w - 3) -^ '-^ ( - ) + 158 MATHEMATICAL FORMUL.^ AND ELLIPTIC FUNCTIONS7.363 I. P Fix^) = (2.i:)"F(")(.v^) + ^li^L^ (2xy-^ F("-')(x2)ax" I n(n - i) (n - 2) (n - 3) ^ n(n - i) (n - 2) {n - 3) (» - 4) in - S) ^,^),,-^ p.-^s^^'>:^ , ^ 2. -y- ^"^ = (2ax)"e°^ I + -77 ^ + 7-/ 57^-^ - Cn-2p{p " l) (log x)^'-'dx" X" ^ „ ^+ Cr^sPiP -l){p- 2) (log X)"-^ - . . . j,where /> is a positive integer. If n

-l)(i>-2) .... 2-1 J-7.369 log (i + x) = Co^p - Ci -^-— + C:/; f I - {p + l){p + 2) - 1 '-'^fP^=^^<^'dx = C. 8.010 Homogeneous equations of the form:dv^ P(x, t)dx - Q(x, 30'where P(x, y) and Q(x, y) are homogeneous functions of x and y of the samedegree. The change of variable: y = vx,gives the solution: 162 DIFFERENTIAL EQUATIONS 1638.011 Equations of the form:dy_ _ a'x + h'y + c'dx~ ax + by + cIf ab' - a'b =j= o, the substitutionX = x' + p, y = y' + q,where ap + bq + c = o,a'p + b'q + c' = o,renders the equation homogeneous, and it may be solved by 8.010.If ab' - a'b = o and b' 4= o, the change of variables to either x and z ox yand z by means of z = ax -\- by, will make the variables separable (8.001). 8.020 Exact differential equations. The equation,P{x,y)dx + Q{x,y)dy = o,is exact ii, ^ _ ^.dx dyThe solution is:fp{:x, y)dx +J I (3(:v, y) - ^ fP{x, y)dx J dy = C,or Jq{x, y)dy +J I P{x, y) - ^^Jq^^, y)dy J dx = C, 8.030 Integrating factors. v{x, y) is an integrating factor ofP(x, y) dx + Q(x, y) dy = o,if |^(^'0)=^(^'^)-8.031 If one only of the functions Px + Qy and Px - Qy is equal to o, thereciprocal of the other is an integrating factor of the differential equation.8.032 Homogeneous equations. If neither Px + Qy nor Px - Qy is equal to o,p . ,(^ is an integrating factor of the equation if it is homogeneous. 164 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS8.033 An equation of the form,P{x, y)ydx + Q{x, y)x dy = o,has an integrating factor: DIFFERENTIAL EQUATIONS 1658.041 The equation can be solved as an algebraic equation in p. It can bewritten (p-R,)ip-R,) . .ip-Rn)=o.The differential equations: p = Ri(x,y),P = ^2(.r, y),may be solved by the previous methods. Write the solutions:/iCv, y, c) = o; /2(.v, J, 6-) = o;where c is the same arbitrary constant in each. The solution of the givendifferential equation is:/i(a-, y, c)UXx, y,c) fn{x, y, c) = o.8.042 The equation can be solved for y:I. y=f{x,p).Differentiate with respect to .t; It may be possible to integrate (2) regarded as an equation in the two variablesX, p, giving a solution3. (j){x, p, c) = o.If p is eliminated between (i) and (3) the result will be the solution of the givenequation.8.043 The equation can be solved for x:I. x=f(y,p).Differentiate with respect to y: If a solution of (2) can be found:3- 4> (>'> P^ = o.Eliminate p between (i) and (3) and the result will be the solution of the givenequation.8.044 The equation does not contain x:f(y,P)=o.It may be solved for p, giving, dv „, .tc = ''(•'•''which can be integrated. l66 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS8.045 The equation does not contain y:fix, p) = o.It may be solved for p, giving, which can be integrated.It may be solved for x, giving, X = F{p),which may be solved by 8.043. 8.050 Equations homogeneous in x and y.General form: f{p,A = o.(a) Solve for p and proceed as in 8.001(b) Solve for -: y = ^m.Differentiate with respect to x:dx _ np)dpX p-fipywhich may be integrated. 8.060 Clairaut's differential equation:I. y = px+f{p),the solution is: y = CX+f(c).The singular solution is obtained by eliminating p between (i) and2. X +f'(p) = O.8.061 The equationI. y = xf(p) + (f>(p).The solution is that of the linear equation of the first order:dx _ fip) ^ ct>'(p)dp p-fip)- p-f(pywhich may be solved by 8.002. EHminating p between (i) and the solution of(2) gives the solution of the given equation. DIFFERENTIAL EQUATIONS 1 678.062 The equation: x4>{p)+y^{p) = x{p),may be reduced to 8.061 by dividing by ^{p). DIFFERENTIAL EQUATIONS OF AN ORDER HIGHER THAN THE FIRST8.100 Linear equations with constant coefficients. General form:T^n + «i T^i + ^2 -i—r6y + • • • • + ^n^ = V{x).dx" dx"^ dx'' 2The complete solution consists of the sum of(a) The complementary function, obtained by solving the equation withV(x) = o, and containing n arbitrary constants, and(b) The particular integral, with no arbitrary constants.8.101 The complementary function. Assume y = e^"". The equation fordetermining X is: X" + cTiX""^ + floX""- + + a„ = o.8.102 If the roots of 8.101 are all real and distinct the complementary functionis: y = cie^'^ + Coe^-'^ + .... + c„g^n^.8.103 For a pair of complex roots:M ± iv,the corresponding terms in the complementary function are: e'^^'O-l cos vx + B cos vx) = Ce^-^ cos {vx - B) = Ce>^^ sin {yx + 6),where C = V.42 + B\ tan^ = |-8.104 If there are r equal real roots the terms in the complementary functioncorresponding to them are:e^'iAi + A.x + Asx^ + .... + A,.x'-^),where X is the repeated root, and Ai, A2, , .4^ are the r arbitrary constants. .8.105 If there are m equal pairs of complex roots the terms in the complementaryfunction corresponding to them are:e'^'l (.4i + A2X + A3X' + .... + A^iX"'-'^) cos vx+ (Bi + Box + Bsx"^ +.... + B^x""-^) sin vx]= e^^lCiCos (vx - 61) + C2X cos (vx - do) + + CmX'"-'^ cos (vx - 6,,,)}= e^'^fCi sin (vx + di) + dx sin (vx + 62) -\- + CmX""-^ sin (vx + ^,„)} 168 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONSwhere X ± i/J, is the repeated root and tan dk BkAk The particular integral.8.110 The operator D stands for — , Z)^ for v",,dx dx^The differential equation 8.100 may be written:(!>" + oi Z)"-i + C2l>"-- + + an)y = f(D)y = V(x)y f{Dyf(D) = (D-\y){D-\,) (D-\n),where X], X2, . . . . . ., X„ are determined as in 8.101. The particular integral is: ^ = gX.x rg(X2-X0x^_^; Ce(^'-^^)'dx Ce~^n^') V{x)dx.8.111 rrjr- may be resolved into partial fractions:f{D) " D - Xi "^ L> - Xo + • • • • + D - X„'The particular integral is:y = Nie^'^ I e-^'='V{x)dx + Noe^-^ I e-^"-^V{x)dx + + NnC^n^ I e-^r^'V {x)dx. THE PARTICULAR INTEGRAL IN SPECIAL CASES8.120 V{x) = const. = c, c8.121 V{x) is a rational integral function of x of the mth degree. Expand ' , , in ascending powers of D, ending with D"". Apply the operators D, D~, , D"^ to each term of V{x) separately and the particular integral will bethe sum of the results of these operations. DIFFERENTIAL EQUATIONS 1 698.122 V{x) = ce'^, unless ^ is a root of f(D) = 0, If ^ is a multiple root of order r of f(D) = ocx^e'' -^where8.123 V{x) = c cos {kx + a).If /I' is not a root of J{D) = o the particular integral is the real part off{ik)If ik is a multiple root of order r of J{D) = o the particular integral is the realpart of CX'e'^^ x+a) where /'^Z^) is obtained by taking the rth derivative oi J{D) with respect to D,and substituting ik for D.8.124 F(x) = c sin {kx + a).If ik is not a root of J{D) = o the particular integral is the real part off(ik)If ik is a multiple root of order r of /(Z)) = o the particular integral is the real .part of 8.125 V(x) = ce'^^-X,where X is any function of x. If X is a rational integral function of x this may be evaluated by the methodof 8.121.8.126 - V(x) = c cos (kx + a)-X,where X is any function of x. The particular integral is the real part ofCel (jS + ik) is formed as in 8.123.8.129 V = cA^'smikx + a).If (/3 4- ik) is not a root oif(D) = o the particular integral is the real part off(^ + ik)If (/3 4- ik) is a multiple root of order r of f{D) = o the particular integral is thereal part of > f^K(3 + ik)8.130 V(x) = x^'X,where X is any function of x.y = x-j^^X + mx- ij-^^jX + ^^y— .T |^.J^jA+The series must be extended to the (m -f i)th term. 8.200 Homogeneous linear equations. General form:d"y , d"~^v , dv , ^^z \dx" dx"~^ ox-Denote the operator: d .X— = U,dxX-^P- = 6(6 - i)(d -2) (d-m + i).dx^The differential equation may be written:F(d)-y= V(x).The complete solution is the sum of the complementary function, obtained bysolving the equation with F(.v) = o, and the particular integral. DIFFERENTIAL EQUATIONS 1718.201 The complementary function.y = CiX^' + C2X^' + + CnX\where Xi, X2, , Xn are the n roots ofF{X) = oif the roots are all distinct.If Xk is a multiple root of order r, the corresponding terms in the comple-mentary function are:x^k{bi + h log .r + ^3 (log xY + .... + ^. (log .v)'-M . If X = )U ± z> is a pair of complex roots, of order r, the corresponding termsin the complementary function are:x-^{[^i + A2 log x + Az (log x)2 + .... + yl, (log xy-^'] cos {v log x)+ [5i + Bn log X + Bz (log xY + .... + 5, (log aO'-i] sin {v log x)].8.202 The particular integral.If F{d) = (^-Xi)(^-X2) (0-X„),y x^^ fx^'--^^-'dx jx^'-^^-'dx fx>''"^''-'-'V(x)dx.8.203 The operator Tym niay be resolved into partial fractions: F{d) - d-\r d-Xo'^ "^ ^ - X„'y = iVix-^' fx-^'-W(x)dx + N2X^'- \ x'^'-'W {x)dx+ + NnX^''fx-^"-W{x)d The particular integral in special cases.8.210 V(x) = cx\ unless ^ is a root of F(d) =0.If ^ is a multiple root of order r of F(d) =0. _ c (log xY^~ F^'\k) 'where F^'^{k) is obtained by taking the rth derivative of F(d) with respect to ^and after differentiation substituting k for d. MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSV(x) = cx'X,I1728.211where X is any function of x F{d + k) X. 8.220 The differential equation:(fl + 5-v)«^ + (a + bx)-'a,£^ + + (a + Ma„-i ^, + a„^ = V{x),may be reduced to the homogeneous linear equation (8.200) by the change ofvariable ,s = a + ox.It may be reduced to a linear equation with constant coefi&cients by thechange of variable:^ e' = a + hx. 8.230 The general linear equation. General form:d^ pd"-'ydx' + Pn + Pnwhere Po, Pi, , P„, V are functions of x only.The complete solution is the sum of:(a) The complementary function, which is the general solution of the equationwith V = o, and containing n arbitrary constants, and(b) The particular integral.8.231 Complementary Function. If yi, yo, . . . . , y„ are n independent solu-tions of 8.230 with V = o, the complementary function isV = Ciyy + C2y2 + + Cnyn.The conditions that yi, vo, . . . . , y„ be n independent solutions is that thedeterminant A 4^ o.A = DIFFERENTIAL EQUATIONS 1738.232 The particular integral. If Ak is the minor of j^:;^ in A, the par-dx'ticular integral is:y = >'i fFA, ^ rvAo ^ rvAn dx.8.233 If >'i is one integral of the equation 8.230 with v = o, the substitutionduy = uyi, V dxwill result in a linear equation of order n - i.8.234 If yi, >'2, ...... yn-i are n - 1 independent integrals of 8.230 withV = o the complete solution is:y = ^y Ckk + cn^ ykj ^' T'^^o'^* dxwhere A is the determinant:A = fi" "Vi (f" "V2 dx"' dx"' d"-'yn-idx"-'d"-^n_,dx"'^ dyidxyi dy2dxy2 dxyn-iand Afc is the minor of d"-dx' in A.SYMBOLIC METHODS8.240 Denote the operators: 8.241 If Z is a function of .r:I dxd_dx (D - m)~^ o = ce""".{$ -m)-'X = x"' jx-(6 — m)~'^ o = ex"". Xdx. Xdx. 174 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS8.242 If F{D) is a polynomial in D,1. F{D)e'"^ = e'»^F(m).2. F{D)e^'X = e"-F(D + m)X.3. e"^'F{D)X = F{D - m)e'"^X.8.243 If F(d) is a polynomial in 6,1. i7(0)^"' = x^F{m).2. F{d)x"^X = x^F{d + W2)X.3. x^F(d)X = Fid - m)x'-X.8.244 x^^ =d(d-i)(d-2) (d- m f i). INTEGRATION IN SERIES8.250 If a linear differential equation can be expressed in the symbolic form;[.v'"F(0)+/(0)]y=o,where F{d) and/(0) are polynomials in 6, the substitution,00 11 = oleads to the equations, Oo/(p) = o,ai^F{p) + ai/(p + m) = o,OiF(p + m) + 02 /(p + 2w) = o,ao:F{p + 2w) + 03 /(p + ^m) = o. 8.251 The equation /(P) = o,is the " indicial equation." If it is satisfied Oo may be chosen arbitrarily, and theother coefficients are then determined.8.252 An equation: ^Fid) + Cl>(d) £;;^y = O,may be reduced to the form 8.250, where,f(d) = ct>(d - m) 6(6-1) (d - 2) (6- m + i).If the degree of the polynomial / is greater than that of F the series always con-verges; if the degree of / is less than that of F the series always diverges. DIFFERENTIAL EQUATIONS 175ORDINARY DIFFERENTIAL EQUATIONS OF SPECIAL TYPES8.300 where X is a function of x only.y = -,—^-T-^ / (.V -0"~^ Tdt + ci-v"-^ + <:2.v"-- + . . . + c„_i x + Cn,where T is the same function of / that X is of x.8.301 dx'' 'where F is a function of y only.If ;/'(v) = 2J F(/y,the solution is: 8.302Put / dv = X + C2.my) + c^}'dyd. Y = 4>{x + c), and this equation may be solved by 8.300.Or the equation can be solved:r dY r dY fYdl -J F{Y)J F{Y) J F (F)'where the integration is to be carried out from right to left and an arbitraryconstant added after each integration. Eliminating F between this result andY = (t>{x + ci)gives the solution.8.303 dx^ \dx''-y' 176 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSPut which may be solved by 8.301. If the sohition can be expressed:Y = (/)(a:),n — 2 integrations will solve the given differential equation.Or putting yP{y) = 2jYdy,r dv r dY r f>'=-7 \c,^yl;{Y)Vj W^^l^{Y)V J W^ dY^{y)\vwhere the integration is to be carried out from right to left and an arbitraryconstant added after each integration. The solution of the given differentialequation is obtained by elimination between this result andY = 4>{x).8.304 Differential equations of the second order in which the independentvariable does not appear. General type: V- ' dx' dxVPut dy dp d^ydx^ dy dx^A differential equation of the first order results: If the solution of this equation is:P-fiy),the solution of the given equation is, 8.305 Differential equations of the second order in which the dependent variabledoes not appear. General type: I'' tx^ d?) = °-Put dy dp ^^^dx' dx dx^ DIFFERENTIAL EQUATIONS 177A differential equation of the first order results:K-^'i^)If the solution of this equation is: the solution of the given equation is:y = C2 + ff{x)dx.8.306 Equations of an order higher than the second in which either the inde-pendent or the dependent variable does not appear. The substitution:dv as in 8.304 and 8.305 will result in an equation of an order less by unity than thegiven equation.8.307 Homogeneous differential equations. If y is assumed to be of dimensions n, X of dimensions i, -^ of dimensions (n - i), -7^, of dimensions (n — 2),dx dx- ^ ^'then if every term has the same dimensions the equation is homogeneous.If the independent variable is changed to d and the dependent variable changedto z by the relations, X = (P, y = ze"^,the resulting equation will be one in which the independent variable does notappear and its order can be lowered by unity by 8.306.If ^', 21 T^. .... are assumed all to be of the same dimensions, and thedx dx-equation is homogeneous, the substitution: will result in an equation in u and x of an order less by unity than the givenequation. 8.310 Exact differential equations. A linear differential equation:^ ^-ly .pdy where P, Po, -Pi, P„ are functions of .r is exact if:dx dx^ ^ ' dx'"- where, 178 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSThe first integral is: e. = -p., ^ dxQn-2 - Fn-2 " ^_, + J^2 ' dP2 d'Ps . , ^„_,rf"-^P«^^1-^^ +^-. •••+(-!)If the first integral is an exact differential equation the process may be con-tinued as long as the coefficients of each successive integral satisfy the conditionof integrability. 8.311 Non-linear differential equations. A non-linear differential equation ofthe Mth order: /d^ d^ dy \d"vto be exact must contain -7^ in the first degree only. Putd"-'y _ d''y dpdx"-'~-^' dx"~dx'Integrate the equation on the assumption that p is the only variable anddp . d"~' y~ its differential coefficient. Let the result be Fi. In V dx - dVi, — f- isdx dx"-'the highest differential coefficient and it occurs in the first degree only. Repeatthis process as often as may be necessary and the first integral of the exact dif-ferential equation will be Vy +V2+ = C.If this process breaks down owing to the appearance of the highest differentialcoefficient in a higher degree than the first the given differential equation wasnot exact. DIFFERENTIAL EQUATIONS 1 798.312 General condition for an exact differential equation. Write:dx dx- ' dx"In order that the differential equation:vi-^', y, y', y", ,3'^"'') = o,be exact it is necessary and sufficient thatdV d /dV\ , d'~ /dV\ , .d-(dV\ 8.400 Linear differential equations of the second order.General form: where P, Q, R are, in general, functions of x.8.401 If a solution of the equation with R = o:y = wcan be found, the complete solution of the given differential equation is:y = cfcv + CIV Ce-fP''- ^ + w fe-^^-^^ ^ fwRe-^^''^ dx.8.402 The general Hnear differential equation of the second order may bereduced to the form: dx^where: y = ve'^-^^''^, 2 dx 48.403 The differential equation:d~v „ dv „S5 + ^S + »' = °'by the change of independent variable toz = fe-fP''^ dx,becomes: d'-y , „ ,^p.By the change of independent variable.dz = Qe-fP^'^ dx, it becomes: d \i dy\ ^ l8o MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS8.404 Resolution of the operator. The differential equation:d-y dvdx- dxmay sometimes be solved by resolving the operator, into the product, rf2 ddx^ dx The solution of the differential equation reduces to the solution ofdv , - CU,r -f- + sy = Cie J PdxThe equations for determining p, r, q, s are:pr = u,drqr + ps + p-^ = v,ds8.410 Variation of parameters. The complete solution of the differentialequation: d'^y dydx^ dxIS y = cUx)^-cMx)+^pR{^)e''^''' {/2(x)/i(0 -/i(-v)/2(a } 4,where Ji{x) and J^i^x) are two particular solutions of the differential equationwith R = o, and are therefore connected by the relation , df2 . df, - ^^-C is an absolute constant depending upon the forms of /i and /2 and may betaken as unity. 8.500 The differential equation:(02 + box) T^ + (ai + bix) -f- + (flo + bfix)y = o.dx- dx8.501 Let D = (ao^i - axb(){aibi - 02^1) - {a^bi - a^b^Y. DIFFERENTIAL EQUATIONS igiSpecial cases.8.502 bo = bi = bo = o.The solution is:where : Xi _ - a I zhVflr - 4000-2X2 2028.503 Z) - o, 62 = o, where: Oi bi . bok = — m = —- A=— — • 02 202 Oi8.504 D = 0, bo:^o: where y = e^Ac, + c, re-(^-+2^)-(o2 + bix)^dx \ , bi oobi — 01^2k = T- m62 '" Z^a'and X is the common root of:a-ih? + OiX + Oo = o,62X- + bx\ + bo = o.8.505 D ::^ o, ^2 = ^1 = o. If 77 = fiO is the complete solution of: where _ 400O2 - Oi- o _ bo \ _ _ Qi402'- 02 2028.510 The differential equation 8.500 under the condition D =}= o can alwaysbe reduced to the form: 8.511 Denote the complete solution of 8.510:8.512 b2 = bi = o: y ^ gXr+CM+t-x)? F{2{fX + VX)"},where: X = _ -^ _ Oi^ - 40n0'2 /4Q2-Y202 402- \gbo-/ ' V902/ l82 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS8.513 b; = o, b,^o: 201where: DIFFERENTIAL EQUATIONS 1 838.521 The function F{p, q, ^) can always be found if it is known for positiveproper fractional values of p and q.8.522 p and q positive improper fractions:f p = m + r, q = n -{- swhere m and u are positive integers and r and s positive proper fractions.F(m + r,n + s, = ( -^)"^.[^-' |^ { ^^ ^r, s, 8 } ]'8.523 p and q both negative:p = - (m - I +r) q = - {n - 1 + s),/r (_ ^ + I _ ,, _ ,, + I - 5, a = ( - i)'" ^+"+^+^-^ 1^ [.-' ^^ { e^ F{s, r, ^ } ]•8.524 p positive, q negative:p = m + r, q = - n + s,F{m + r, - n + ., ^) =^ \^^n+r-r-s^ ^(^ _ ,^ i _ ,^ t)J . 8.525 ^ negative, q positive:p -= — m + r, q = n + s,/7(_^ + ,,n + ,,^) = (_i)-+»e-?|;-|^^-+i— ^^|,^^(I-,,I-r,0}]• 8.530 If either p or q is zero the relation Z) = o is satisfied and the completesolution of the differential equation is given in 8.502, 3.8.531 li p = m, a positive integer: = F{ni, q, ^) = ^1^ [^^^-'/^-^^^'^^] + ^^^[^'^"]•8.532 li p = m, a positive integer and both q and ^ are positive: (f) = F{m, q, = d I w'"~^(i - uy-^e-^'' du + c^e'^ / (^ + u)"'-^ u'^'^ e~^" du.8.533 If g = «, a positive integer:(^ = F{p, n, = c,e-i |^l[^' '\f^'-' '-^ ^^] + ^^^"^ |iJ^ [^^^']•8.534 If g = w, a positive integer and both p and ^ are positive:4) = F(p, n, ^) = Ci I mp-i(i - «)"-! e-^" du + C2e-^ I {i + u)p-^ u"'^ e"'" Jw.t/o I/O l84 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS8.540 The general solution of equation 8.510 may be written:(j) = F(p, q, ^) = CiM + CiN,Jo 9> oJo K>o i^jf _ IM£M / . _ ^ i ^ P^P + ^) S. _ p(p + ^)(p + 2) ^ ^ \^- T{s) 1' "sir s{s +i)2l s{s+i){s + 2) 3\ + /s =^ p + q,T(q)e-^ ( (p - i)q (p - i)(p - 2)q(q + i)(p- i)(p - 2) {p -n- i)(q)(q + i) (q + n - 2)p(p- i)(p- 2) . . . (p- n)q{q + i){q + 2) . . . . {q + n - i) \"^ «!^" /'where o < p < i and the real part of ^ is positive. THE COMPLETE SOLUTION OF EQUATION 8.510 IN SPECIAL CASES8.550 p>o, q>o, real part of ^>o:F(p, q, 9 = ci / zo, q>o, ^o:F{p,g,0 = ^^~^~'' \ci I {i-u)-Pu-'>e-^''du + Coe-^ j m-p(i +w)-9e-^"^M l8.553 po, qo: F(i - r, I - q, ^) = ci I u~''(i - u)~'^e~^''du+ €26-^ I (i + M)-''M-'g-^" Jm, ^o,q = n + s, where n is a positive integer and s a proper fraction.F(p, n + s,0 = e-'^^ I e^^'-^-^F(i - ,, i - ^, ^) 1 , ^>o: F(i - s, I - p, ^) =- Ci I u-'{i - u)-Pe-^"du+ C2e-^ I (i + u)-'u-Pe-^''du, ^- =- o,y = e-(p+«"')^il,/(Q:, -, -q{x - m)A.8.631 8.632g + 2(i> + ?x) ^ + I 9 + c(i - 4«) + {p + qx)' - cKx - my- 1 y = o,y = g-px-iqx^-Mx-my- m((x^ l^ c{x - m)A •8.633 7-gy^g-(p+Ox^ 2 ^/ (q;^ Y, 2/0-).8.634d^y ( 2y - 1 ^ \ 1 ^y+ j ^ '^ ~ ^^ + (a- + 267 - 4Q:f) + 2a{b - c)x + b{b - 2c)x" \ y = o,y = 6-"='-'^='' M(a, 7, cx^).8.635 ip"^ - f)x^' + r{pq + 7/ - 2a/).r'- + - r'^{y - q){2 - q - y) \y = oI 4 Jy = e ' ;^^-«^T7/ 2/aA.1-^ il/(^a, 7, — j- 8.640 Tables and graphs of the function M(a, 7, x) are given by Webb andAirey (Phil. Mag. 36, p, 129, 191 8) for getting approximate numerical solu- 188 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONStions of any of these differential equations. The range in x is i to lo; ina, +0.5 to +4.0 and -0.5 to -3.0; in 7, i to 7. For negative values of x theequations of 8.61 may be used. SPECIAL DIFFERENTIAL EQUATIONS8.700 where X{x) is any function of x. The complete solution is: 8.701 g,.g, „,,..-(.,The complete solution, satisfying the conditions:X = o y = yo,dy , y = g-i«^ < yo where8.702 ^. = yo sin 8.703 8.704 n'x I , K . ,\ -, h yo I cos nx H ; sm w x\+ -, A-i-^^^-^) sin n' {x - %)X (0 di,^ Jo ' J " "' ^~ J fe-J'^^^^'^g{x)dx + c,^'^' eff(y)'^y dy[c,-2fe'ff^y^'yg{y)dyY^ + C2. dx''^- B(y)dy dy^ +C2.g{y)dyj(^y-^ dy 8.705 8.706 8.707 8.708 DIFFERENTIAL EQUATIONS 189 y = e-^^jci + c fx-^'e^' dx] d'^y a dy b _dx^ X dx x'^ I. (a-iy>4b; X = - V{a - i)- - 4b _ O— I 3' = X ~2~ {CiX + C2X~^}-2. (a - iy<4b; X = - V46 - (a - 1)22a— I y = .-v; 2 {ci cos (X log n-) + C2 sin (X log x)}'3. (a - 1)2 = 46 a— I y = x~"r(ci + C2 logs-).8.709 g+,,,|+(, + ,,,., = o. I. a<6, X = \/b — a,y = e 2 (cieV + C2e~^^).2. a>&, X = Va - b 8.710 _ 6^2y = e 2 (ci cos Xx + C2 sin \x). y = c:i(a + bx) + Co j e-^ - (fl + ^a;) / -r^ e-^' dx \ igO MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS8.711 8.712 8.713 ^ 2 ^ 2 _ ^if .ay = - < 1 cos ux" + Co sin ux + — 5 •^ a; [ "^ IJ?d*y jd^y d-y , , <^3' ,y = cie-P'^'lpi sin (coi-r + ai) + coi cos (wix + ai)}+ C2e~''''' { P2 sin (CO2.V + a^) + CO2 cos (a;2.r + ai)},where: 40)1^ = S + C-2C?"+ 2\/s^ - 4a - 2 ^+ 2)(w + 3)(n + 4) ^_(„^,^ _^ I ^2-4-(2« + 3)(2«+ 5) ... .J9.011P: „(cos P) = (-i)" ;, ,, ^ ^1 sm2" - ^-p- sm^''-^ cos^ d(2IO!9.012 ,(2«+ l)! [^._,„ ^ _^ ^ (2W)23!-f2n+i (cos ^) = (-1)" „ , ' \ sin-" 6 cos 6 - -—/- sin-"~- 6 cos^2"^"(«!)'^ (2W+1)! J(Brodetsky: Mess, of Math. 42, p. 65, 191 2)191 192 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 02 Recurrence formulae for Pn(x):{n + i)P„+i + nPn-i = (2» + i)xPn.dPn+l dPn-1{in + i)Pin + i)P„nP dx dxdPn+i dPndxdPn _ dPn-1dx dx(i-x'^)^=(n+i)(xPn-Pn+i).{l-X^)^ = n{Pn-l-xPn).{2n + l)(l - X^)^ = «(« + l)(P„_l - Pn+x). ,028 Recurrence formulae for Qn{:x). These are the same as those for Pn{x). 9.030 Special Values.Po{x) = 1, 9.031 Pi(x) = X,P2(x)=U3x'-i),Pb(x) = kiSx' - 3x),P,(x) = H3Sx'-30x' + 3),P^(x) =i(63x5-7o^'+i5-'^-),Pe(x) = iV(23i^' - 315^-' + 105-^"' - 5),Py(x) = 1^(429-^'^ - 693^^ + 315^^-35^),Ps{x) = Tis{(>43Sx^ - i2oi2:r« + 693o:k;4 - 1 260x2 + 35). , . I x+ 1Qo(x) = -^\og^--^, Q2{x) = lP.ix)\0g^^-lx,Q,(,)=lp,(,)log^^-lx^-+'-- DIFFERENTIAL EQUATIONS 1 939.032 ^^271+1(0) = o,Pn{l) = I, .^(--'^0 = {-^YPn{x).9.033 li z = r cos d:dz r \ J^^/^P^,(cos«)-P,..,(cose)]9.034 Rodrigues' Formula:Pn{x) =-^,fl (^' - l)"-2"w! ax"9.035 If z = r cos ^: P„(cos0)=^%"+^-^f^9.036 If w ^ w : where: ^ _ I-3-5 .... (2r- i) MEHLER S INTEGRALS9.040 For all values of n: 2 f9 COS (n + h)(j)d(j)P„(C0S d) = — I , r J. a\^Jo a/2 (cos (/) - COS &)9.041 If n is a positive integer: 2 /''^ sin (w + \^^d^^^ V 2 (cos d - COS x> + i, two solutions of this differentialequation are the associated Legendre functions o:w = (I-.^•'F^•9.071 If «, w, r are positive integers, and n>m, r>m:jy pZ (.r) p7 (.v) dx = oiirdf:n,2 (n + m) ! ..= 7 TT li r = n.211 + I {n — ni)l 9.100 Bessel's Differential Equation(Py T- dydx^ X dxd^ i / v~\9.101 One solution is: 002^J'+2fc 9.102 A second independent solution when p is not an integer is:y = J-Ax).9.103 li V = n, an integer:J_n{x) = {-iyJn(x).9.104 A second independent solution when v = n, an integer, is:Tw N 7 / N 1 X X^' (w ^ ^ - i)! /aA^^-ttI „(x) = 2/. (.r)- log- - 2j YiS (-')"Wi^!©""' I ^(* + -^ + '^(^ + « + "(see 6.61). DIFFERENTIAL EQUATIONS I979.105 For all values of v, whether integral or not:Yp(x) = -r^— (cos virJpix) - J-v{x)],^ ^ sin I'TT V /J-p{x) = cos vtJu(x) - sin vtYp{x),Y-p{x) = sin virJv{x) + cos VTrYp{x).9.106 For V = n, an integer:Y.n(x) = (-i)"F„W.9.107 Cylinder Functions of the third kind, solutions of Bessel's differentialequation: Hr (x) = Ju{x) + iYy{x).2. Hy (x) = J„{x) - iYp(x).3. H-, (x) = e"""' hI(x).4. H-\ (x) = e-'"''Hlix).9.110 Recurrence formulae satisfied by the functions /^, Yp, Hp, H p, Cprepresents any one of these functions. I. Cp-i(x) - Cp+i(x) = 2 — Cpix). 2. 2j/C _i(.v) + Cp+i(x) = — Cp{x). 3. ^Cp (x) = C_i (x) - - Cp (x) . 4. -^C (x) = ^Cp{x) - Cp+x{x).ix'CpiA =x^Cp-i(x).d_dx^^^ = ^ I Cp+,{x) + C._2(x) - 2Cp{x) I5-6.9.111I. Jp{x)^^ - Yp{x)^^ = — • 2. /.+i(.r)r.(.r) - /.Cr)F.+i(A;) = — • ax ax TTX TTXASYMPTOTIC EXPANSIONS FOR LARGE VALUES OF A:9.120 _1. Jp(x) = y ^, I P (x) cos (x - ^-^^^^ tA - Qp{x) sin Cx - ^^^^ 7r) | 2. Yp{x) = v/— s Pp{x) sin ix tt j + Qp(x) cos Lv tt 198 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS hUx) = e ''(' "^ '"V^- r "^"^^ ~ '"whereP„(,v) = X +2 (-')' ^^^^^^^«^^'' (4^'- 4* - O(2/^)! 26^a;2*o r^^ -V r 1)^+1 (4^- - I-) (4^- - 3-) (4^^ - 4^ - 3 ) SPECIAL VALUES9.130 +w(^+^^)(r 7 = 0.5772157 (6.602).9.131 Limiting values for x = o:Jo(x) = I,Ji{x) = o,Fo(:.) = ^(log^+7),Y,{x) = -—.TX 33-4+ -^ Mx) 1J DIFFERENTIAL EQUATIONS " IQQ9.132 Limiting values for x = co : sin I .V — —Fo(x) = "^fjx) = 200 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS9.162 /,yVir(.+i) -/<=If « is an integer:9.153 J2n{x) = — I COS {X sin (/)) COS {2n(f))d4) = - I 2 •9.154 I cos [X COS 0) COS {27l(p)d(p = I ^ •9.155 J,n(x) ^J2n+l{x) = J2n+l(x) = Jn(x) 1 f"" 2 f-— / sin (x sin 0) sin (2« + i) (})d4> = — / ^9.156 ^^—^ / sin {x cos 0) COS (2» + i)(j)d4> = ^ / ^ •T^ Jo 'K Jo9.157 I /'+'' T r'''= / g-in0+« sin <^ JA = I g-in+ixsm(f> ^(k^2'KJ--k 27rJoINTEGRAL PROPERTIES9.160 If Cv(^i.t) is any one of the particular integrals:/.(mx), F.(mx), i7i(Mx), iyt'(M^-),of the differential equation:d^y ^ \dy I ^ v\d?+xd-x+[^-x^)y = '''rI C„{iJikx)Cv{iJ,ix)xdx ^ M;.2 - u^2 1 ^ { f^iCu([Xkx)Cp'{^iix) - tXkCp{iJ.ix)Cy'{iXkx) I ,iXk^iii.9.161 If /Xa: and ixi are two different roots ofCv{lJ.b) = o,j Cv{likx)C,{iXix)x dx = /^ , \ fXkC,(fx,a)Cj{tXka) - jiiCv{iik.a)Cj {^iio) 1 •9.162 If fjLk and ijli are two different roots ofCu'ina) I^ ?^7—T = Pi^ + g-,and Ci;(/x&) = o,/ Cv{iJ.kx)Cv(iJiix)xdx = pC^.{fXka)Cp{iJiia).If )Ufc = jU;:J C,{iXkx)Cv{iiix)xdx = i I b'C'HiJLkb) - a'C^iiXua) - U - —}jCv''{lXka) |. DIFFERENTIAL EQUATIONSEXPANSIONS IN BESSEL S FUNCTIONS9.170 Schlomilch's Expansion. Any function /(:v) which has a continuousdifferential coefficient for all values of x in the closed range (o, tt) may be expandedin the series: f{x) = Co +^ akJo{kx),k=i/IT pit:u i ^ f {u sin d)dddu, ak = — I ucosku I 'f'(u sin 6) dddu. COfix) = Oo-X" + X V OkJniOikX) 0J^{iikh) = o,J^ xj{x)J^{\Xkx)dx - pf{a)Jo{iXka)bUo'~{fikb) - aUo'-'ifJiko) - (a^ + 2p)J^KiJ-ka)(Stephenson, Phil. Mag. 14, p. 547, 1907) 9.180 SPECIAL EXPANSIONS IN BESSEL'S FUNCTIONS 1. sinx = 2^ {-iYJ2k + i{x),k = o CO2. cos X = /o(.t) + 2 /j i-iyJikix). 202 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS9.181 001. COS (x sin 6) = Jo{x) + 2^^ J2k(x) cos 2k6,k = I002. sin {x sin 6) = 2^^J2k^-\{x) sin {2k + i)^. /« + 2t(x), k = o9.183^^ ={ .og| - ^(. H- .)}/(.) .2 (-:)-i;±^.„..W* = I . /.W log - -2, (- I)' ,-rT(. + . + .) y (see 6.61) DIFFERENTIAL EQUATIONS9.202 Special values.Ji(x) = y — sin x,T TTX 9.203 9.204 9.205 J (x) = V ^ (—-^ - cos XT TTX \ X /-!(*) -V^cosx fl>; (X) = i\/Te-^', 204 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS 9.210 The differential equation:(Py 1 dv / , v^\ with the substitution, X = iz,becomes Bessel's equation.9.211 Two independent solutions of 9.210 are:h (x) = i-" J^ (ix),K^ {x) = e^""' ^ h\ {ix).9.212 If V = n, an integer: k =0Kr. (x) = i"+i ^ Hi (x).9.213 ^'' (-^^ = ~7^r ^ (~T I cosh (x cos (^) sin^'' (f)d(f), ii:. (x) r(^ "^V (')'' / ^^^^^^ *^^~' '•''''•^ ^<^.9.214 If :j; is large, to a first approximation:/„ (x) = (27rx cosh /3)-^ e^ (^^^ ^ - ^ ^i"h /3)^ii:„ (x) = 7r(27r:C cosh 13)-^-' (cosh ^ - ^ sinh /3)^w = x sinh /3.9.215 Ber and Bei Functions.ber X + ihe'ix = I (xVi),ber x — iheix = Io(ix\/i),ber X = I - ^^ 1 - ) +(2 0^2/ (40^2 DIFFERENTIAL EQUATIONS 2059.216 Ker and Kei Functions:ker X + i kei x = Ko(x\/i),ker X - i kei x = Ko{ix\/i),ker X = ( log I - t) ber ,v + j bei .v -^ (: + i) Q' kei..= (log?-7)bei.v-^ber. + gJ-^(,+i + i)0J + ., 9.220 The Bessel-Clifford Differential Equation:d-y , . , \ ^^' ,ax- axWith the substitution: s = x^/'y u = 2\/x,the differential equation reduces to Bessel's equation.9.221 Two independent solutions of 9.220 are:COC.(a-) = x'U, (iVx) = y, (-1)^LJ^ ' /^!r(t' + yfe + i)'* = oDv{x) = x'l Yv{2yjx).9.222 xCvvi^y^) = (j^ + 1)0+1W - C(x).9.223 If j^ = «, an integer:Cn{x) = {-iY~C^{:x),ox" 9.224 Changing the sign of v, the corresponding solution of:d-y , , dy ,y = x^Cvix). j206 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS9.225 If V is half an odd integer:sin {2\/x + e)QC^-) = 2\/xr ( -\ - A.r ( A - sii^ (2\/x + e) _ cos (2\/x + e)a.v 4.V- 2.V ^ , V ^ ^ / N 3-4^ . . /- , X 3 COS (aV.T + e) C_i(x) = -cos (2\G + e),C-i(x) = a;'C|(a;),C_|(^) = x'C^ix). € is arbitrary so as to give a second arbitrary constant.9.226 For x negative, the solution of the equation: when V is half an odd integer, is obtained from the values in 9.225 by changingsin and cos to sinh and cosh respectively.9.227(W + n + l) / Cm+l{x)Cn+l{x) dx = - xCm+\{x)Cn+\{x) - Cm{x)Cn{x),(W + W + l) / X'"+"CmCT))Cn{x) dx = .T^+^+l xC„^+,(x)Cn+l{x) + C„.(a-)C„(.t)9.228I. / C-x{x cos^ (f)) d(f) = TvCoix). 2. / Ci(xcos-(/)) d4) = 7rCi(a-). 3. I Co(x sin^ (}>) sin (f) d4> = Ci(x). 4. I Ci(,T sin- (^) sin^ cf) d(f) = C](.t).r^ / • , X • . 7 , I - cos 2\/xCi{x snr 9) sin (p dcp = ^ DIFFERENTIAL EQUATIONS 207,9.229 Many differential equations can be solved in a simpler form by the useof the Cn functions than by the use of Bessel's functions.(Greenhill, Phil, Mag. ^8, p. 501, 1919) 9.240 The differential equation:d'^y 2(n + i) dydx^ X dxwith the change of variable: y = zx~^~^,becomes Bessel's equation 9.200.9.241 Solutions of 9.240 are:1. y = o;-"-^ Jn+\{x).2. y = a;-"-* Yn+\{.x).3- 3; = x-»-^ ^;,+iW-4. y = o;-"-^ Wl+iix).9.242 The change of variable: X = 2\/z,transforms equation 9.240 into the Bessel-CHfford differential equation 9.220.This leads to a general solution of 9.240:y = Cn^i (^).When n is an integer the equations of 9.225 may be employed.(x^\ sin {x + e)4/ ^2 sin (x + e) cos {x + e)^)9.243 The solution of d'^y 2(n + i) dvmay be obtained from 9.242 by writing sinh and cosh for sin and cosrespectively.9.244 The differential equation 9.240 is also satisfied by the two independentfunctions (when n is an integer): , / s I I c?\" sin X I-3-5. .(2»+.)2/^ '' Jlh• . , 2« + l) Lmd 2^k ! (2« + 3) (2W + 2^ + l) 2o8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS I d \" cos X ^ ^ ' xdx X ^ I-3-5- • •(2»-i)y ._., ^:v-" + ^ ^J 2^'^!(i - 2n) (3 - 2n) .... {2k - 2n - 1)k =9.245 The general solution of 9.240 may be written: _ /i d_y Ae^ + Be-^""X dx,9.246 Another particular solution of 9.240 is: in^-ix J- ^ ^ ;j(^ _^ i) ^ (.„, _ i)n(n + i) (w, + 2) I-2-3/nCr) = ^^^ I + " + ' -^—zTTT — -^ +2-4-6 . . . . 2n{ix)" J9.247 The functions xpnix), ^„(x),/„(a') satisfy the same recurrence formulae: 9.260 The differential equation:d?-y n(n + i)it.---^y + y-°'with the change of variable: y = u\/'Xis transformed into Bessel's equation of order n + -.9.261 Solutions of 9.260 are: I. '" ' » 2 " ^ ' \ x ax/ I ^ \" cos x2. 1 ' " " Xc.(.r).(-i)..v/T-^-"-'.W = »=""(- sl-)X dxj X9.262 The functions Sn{x), Cn(x), En(x) satisfy the same recurrence formulae:dSn(x) w + I e / \ c r \I. , = Sn{x) - Sn+l{x).dx x DIFFERENTIAL EQUATIONS 209 3. Sn+l{x) = ^— 6'„(.V) - Sn-l{x). 9.30 The hypergeometric differential equation:x(i-x)^,+ I 7 - (a + /3 + i).r \-£- al3y = o. 9.31 The equation 9.30 is satisfied by the hypergeometric series:f («, ,3, T,.v) = .+g^g.v+ "<"+'>^f +> .'17 1-2 7(7 + l) I •2-3 7(7 + i) (7+ 2)"^The series converges absolutely when .ti. WhenX = +1 it converges only when a + ^ - 7o.9.421 9.422fi = op^i = o 9.423 '-' 3 9.424I /" ^ i Virtph = — ^ 5 K TT I-3-5 . . . {2n+ i)V TV r r(i + V)'where v may have any real value, except a negative integer. (Conjectural.)9.425 p — a ' I = - (g-^' - i)p — a a9.426 With p = aq"^, g2n+lj_ = ( _ i)^2n+l, -l)n I-3-5 • • • (2^-1)(2aO'V7raf MATHEMATICAL FORMUL.^ AND ELLIPTIC FUNCTIONS9.427 qe- y/irat e~9.428 If z = —^,2^at s: "'dvy/irI X /•°° , dv9.43 Many examples of the use of this method are given by Heaviside: Electro-magnetic Theory, Vol. II. Bromwich, Proceedings Cambridge PhilosophicalSociety, XX, p. 411, 192 1, has justified its application by the method of contourintegration and apphed it to the solution of a problem in the conduction of heat.9.431 Herlitz, Arkiv for Matematik, Astronomi och Fysik, XIV, 1919, hasshown that the same methods may be applied to the more general partialdifferential equations of the type, and the relations of 9.42 are valid.9.44 Heaviside's Expansion Theorem.The operational solution of the differential equation of 9.41, or the moregeneral equation, 9.431, satisfying the given boundary conditions, may bewritten in the form, F(p) where F(p) and A(p) are known functions of p = ^- Then Heaviside'sdtExpansion Theorem is: ; F(o) , ^ F(a)A(o) ^JaM{OL) jwhere a is any root, except o, of A(/>) = o, A'(/>) denotes the first derivative ofA(/>) with respect to />, and the summation is to be taken over all the roots ofA(/>) = o. This solution reduces to w = o at i = o.Many applications of this expansion theorem are given by Heaviside,Electromagnetic Theory, II, and III; Electrical Papers, Vol. II. HerHtz, 9.431,has also applied this expansion theorem to the solution of the problem of thedistribution of magnetic induction in cyUnders and plates.9.45 Bromwich's Expansion Theorem. Bromwich has extended Heaviside'sExpansion Theorem as follows. If the operational solution of the partialdifferential equation of 9.41, obtained to satisfy the boundary conditions, is DIFFERENTIAL EQUATIONS 213where C is a constant, then the solution of the differential equation is r ^ F{a)G { Not + Ni + 2ja'where A^o and iVi are defined by the expansion,^^=No + N,p + N.y' + ...;a is any root of A(/>) = o, A'(p) is the first derivative of A(p) with respect to p,and the summation is over all the roots, a. This solution reduces to « = o at/ = o. Phil. Mag. 37, p. 407, 1919; Proceedings London Mathematical Society,15, p. 401, 1916.9.9 References to Bessel Functions.Nielsen: Handbuch der Theorie der Cylinder Funktionen.Leipzig, 1904.The notation and definitions given by Nielsen have been adopted in the pres-ent collection of formulae. The only difference is that Nielsen uses an upperindex, J"(x), to denote the order, where the more usual custom of writing /„(.v)is here employed. In place of Fi" and 112" used by Nielsen for the cyHnderfunctions of the third kind, Hj and Hj^ are employed in this collection.Gray and Mathews: Treatise on Bessel Functions.London, 1895.'The Bessel Function of the second kind, F„(x), employed by Gray andMathews is the function J Vn{x) + (log 2 - y)Jn(x),of Nielsen. SchafheitUn: Die Theorie der Besselschen Funktionen.Leipzig, 1908.Schafheitlin defines the function of the second kind, F„(a-), in the same wayas Nielsen, except that its sign is changed.Note. A Treatise on the Theory of Bessel Functions, by G. N. Watson, CambridgeUniversity Press, 1922, has been brought out while this volume is in press. This Treatise givesby far the most complete account of the theory and properties of Bessel Functions that exists,and should become the standard work on the subject with respect to notation. A particularlyvaluable feature is the Collection of Tables of Bessel Functions at the end of the volume andthe Bibliography, giving references to all the important works on the subject.9.91 Tables of Legendre, Bessel and alhed functions.Pn(x) (9.001). ^ A second edition of Gray and Mathews' Treatise, prepared by A. Gray and T. M.MacRobert, has been published (1922) while this volume is in press. The notation of the firstedition has been altered in some respects. 214 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSB. A. Report, 1879, pp. 54-57. Integral values of n from i to 7 ; from x = o.oito X = 1.00, interval o.oi, 16 decimal places.Jahnke and Emde: Funktionentafeln, p. 83; same to 4 decimal places.P„(cos 6)Phil. Trans. Roy. Soc. London, 203, p. 100, 1904. Integral values of n fromI to 20, from 6 = o to 6 = go, interval 5, 7 decimal places.Phil. Mag. 32, p. 512, 1891. Integral values of n from i to 7, = o to6 = 90, interval i; 4 decimal places. Reproduced in Jahnke and Emde, p. 85.Tallquist, Acta Soc. Sc. Fennicae, Helsingfors, 33, pp. 1-8. Integral valuesof n from i to 8; 6 = o to 6 = go, interval i, 10 decimal places.Airey, Proc. Roy. Soc. London, 96, p. i, 1919. Tables by means of whichzonal harmonics of high order may be calculated.Lodge, Phil. Trans. Roy. Soc. London, 203, 1904, p. 87. Integral values ofn from i to 20; 6 = o to 6 = go, interval 5, 7 decimal places. Reprinted inRayleigh, Collected Works, Volume V, p. 162.dPnjcos 6)ddFarr, Proc. Roy. Soc. London, 64, 199, 1899. Integral values of n from i to 7;= o to = 90, interval i, 4 decimal places. Reproduced in Jahnke and Emde,p. 88.Joix), Mx) (9.101).Meissel's tables, x = o.oi to x = 15.50, interval o.oi, to 12 decimal places,are given in Table I of Gray and Mathews' Treatise on Bessel's Functions.Aldis, Proc. Roy. Soc. London 66, 40, 1900. x = o.i to x = 6.0, interval0.1, 21 decimal places.Jahnke and Emde, Funktionentafeln, Table III. x = o.oi to a; = 15.50,interval o.oi, 4 decimal places.Jn{x) (9.101).Gray and Mathews, Table II. Integral values of n from n = o to n = 60;integral values of x from x = i to x = 24, 18 decimal places.Jahnke and Emde, Table XXIII, same, to 4 significant figures.B. A. Report, 1915, p. 29; n = o to n = 13.X = 0.2 to X = 6.0 interval 0.2 6 decimal places,X = 6.0 to X = 16.0 interval 0.5 10 decimal places.Hague, Proc. London Physical Soc. 29, 211, 1916-17, gives graphs of /„(x)for integral values of n from o to 12, and n = 18, x ranging from o to 17. - J Fo(x) = Go{x); - J 7i(.r) = Gi(.t).B. A. Report, 1913, pp. 1 16-130. x = o.oi to x = 16.0, interval o.oi, 7decimal places. J DIFFERENTIAL EQUATIONS 215B. A. Report, 1915, :*; = 6.5 to x = 15.5, interval 0.5, 10 decimal places.Aldis, Proc. Roy. Soc. London, 66, 40, 1900: x = o.i to x = 6.0. Interval0.1, 21 decimal places.Jahnke and Emde, Tables VII and VIII, functions denoted Ko(.v) and Ki(.r),X = 0.1 to X = 6.0, interval o.i; x = o.oi to x = 0.99, interval o.oi; x = i.oto :« = 10.3, interval 0.1; 4 decimal places. - ^ F„(x) = Gn(x).2B. A. Report, 1914, p. 83. Integral values of n from o to 13. x = o.oi toX = 6.0, interval 0.1; x = 6.0 to x = 16.0, interval 0.5; 5 decimal places. - \\{x) + (log 2 - t)/„(.v). Denoted Vo(x) and l\{x) - Yi{x) + (log 2 - 'y)Ji{x). respectively in the tables.B. A. Report, 1914, P- 76, x = 0.02 to x = 15.50, interval 0.02, 6 decimalplaces.B. A. Report, 1915, p. 33, x = 0.1 to x = 6.0, interval 0.1; x = 6.0 toX = 15.5, interval 0.5, 10 decimal places.Jahnke and Emde, Table VI, x = o.oi to x = i.oo, interval o.oi; x =1.0to X = 10.2, interval 0.1, 4 decimal places.Yo{x), Fi(x). Denoted No(x) and Ni(x) respectively.Jahnke and Emde, Table DC, .r = 0.1 to x = 10.2, interval 0.1, 4 decimalplaces. - Yn{x) + (log 2 - 7) J,Xx). Denoted Y„(x) in tables.B. A. Report, 191 5. Integral values of n from i to 13. x = 0.2 to x = 6.0,interval 0.2; .v = 6.0 to x = 15.5, interval 0.5, 6 decimal places.J. + ii-^).Jahnke and Emde, Table II. Integral values of ;/ from « = o to n = 6, andw = -I to « = -7; .T = o to X = 50, interval i.o, 4 figures.J.(x), /_.(x).Watson, Proc. Roy. Soc. London, 94, 204, 1918.X = 0.05 to X = 2.00 interval 0.05,X = 2.0 to X = 8.0 interval 0.2,4 decimal places./«(«), Ja-l{0i)TV TVYa{oi), Ya-i{a). Denoted Ga{a) and Ga-i{a) respectively. 2l6 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSJ Ya{a) + (log 2 - y)Jaia), - Ya-i{a) + (log 2 - y)Ja-i(a). Denoted -Ya{a) and -Ya-i(a).Tables of these six functions are given in the B. A. Report, 1916, as follows:From a DIFFERENTIAL EQUATIONS 21 7Aldis, Proc. Roy. Soc. London, 66, 142, 1900; x = o.i to x = 6.0, interval0.1, 21 decimal places.Jahnke and Emde, Tables XV and XVI, same range, to 4 places.Jo{xVi)-Gray and Mathews, Table IV; x = 0.2 to x = 6.0, interval 0.2, 9 decimalplaces.Yo{xV~i) (9.104) Denoted NoixVi) in table.Hl(xVt), H\{xVi).Jahnke and Emde, Tables XVII and XVIII; x = 0.2 to x = 6.0, interval0.2, 4-7 figures. '- Hliix) = K,ix), (9.212). -^//J(fx) =K,(x),Aldis, Proc. Roy. Soc. London, 64, 219-223, 1899; x = o.i to .v = 12.0,interval 0.1, 21 decimal places.Jahnke and Emde, Table XIV; same, to 4 places.iHl{ix), -Hliix) (9.107).Jahnke and Emde, Table XIII; x = 0.12 to ::c = 6.0, interval 0.2, 4 figures.ber. bar'., ^^^ifi).bei X, bei x,B. A. Report, 1912; a; = 0.1 to re = lo.o, interval 0.1, 9 decimal places.Jahnke and Emde, Table XX; a; = 0.5 to x = 6.0, interval 0.5, and x = 8,10, 15, 20, 4 decimal places.ker X, ker' x, , . , ., (9.216).kei X, kei x,B. A. Report, 191 5; x = 0.1 to x = lo.o, interval 0.1, 7-10 decimal places,ber^ X + bei^ x,ber'2 X + bei'2 x,ber X bei' x — bei :jj ber' x, and the corresponding ker and keiber X ber' x + bei x bei' x, functions.B. A. Report, 1916; x = 0.2 to x = lo.o, interval 0.2, decimal places.Sn{x), S'nix), logSM, log^'.Cr),C„(.v) , C'„(.r) , log C„(.r) , log C'„ (x) , (9.261) .£„(x), £'„(x), log£„(.v), log£'„(.v),B. A. Report, 1916; integral values of n from o to 10, x = i.i to x = 1.9,interval 0.1, 7 decimal places. 2l8 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS ^w = 7in(-i)^'-'-=(!)- t:^/''-^5407 * V2Table I of Jahnke and Emde gives these two functions to 3 decimal placesfor X = 0.2 to X = 8.0, interval 0.2, and x = 8.0 to x = 12.0, interval i.o.Roots of Jo(x) = o.Airey, Phil. Mag. 36, p. 241, 1918: First 40 roots (p) with correspondingvalues of /i(p), 7 decimal places.Jahnke and Emde, Table IV, same, to 4 decimal places.Roots of Ji(x) = o.Gray and Mathews, Table III, first 50 roots, with corresponding valuesoi Jo(x), 16 decimal places.Airey, Phil. Mag. 36, p. 241: First 40 roots (r) with corresponding valuesof Joir), 7 decimal places.Jahnke and Emde, Table IV, same, to 4 decimal places.Roots of /„(.t) = o,B. A. Report, 191 7, first 10 roots, to 6 figures, for the following integralvalues of n: o— 10, 15, 20, 30, 40, 50, 75, 100, 200, 300, 400, 500, 750, 1000.Jahnke and Emde, Table XXII, first 9 roots, 3 decimal places, integralvalues of n 0—9.Roots of: 7r(log 2 — y)Jn(x) H F„(x) = o. Denoted F„(x') = o in table.Airey: Proc. London Phys. Soc. 23, p. 219, 1910-11. First 40 roots forn = o, I, 2, 5 decimal places.Jahnke and Emde, Table X, first 4 roots for n = o, 1. E decimal places.Roots of:J^l ~ °' Denoted No{x) and Ni(x) in tables.Yi{x) = o.Airey: 1. c. First 10 roots, 5 decimal places.Roots of:Jo{x) ± (log 2 - 7)/o(.t) + - Yo(x) = o. Denoted Jo(x) ± Yo{x) = o./i(.t) + (log 2 - y)Ji(x) + - Yi(x) = o. Denoted Ji{x) + Yi{x) = o.2Jo{x) - 2 (log 2 - 7)/o(-^") H Yo(x) = o. Denoted Jo(x) - 2Yo(x) = o.2io/o(x) ± (log 2 - y)jQ{x) -\— Fo(x) = o. Denoted io/o(-^") ± Yq{x) = o.2 DIFFERENTIAL EQUATIONS 219Airey, 1. c. First 10 roots, 5 decimal places.Roots of- Jn{:x) ^ In{x) °'Airey, 1. c. First 10 roots: n = o, 4 decimal places, « = i, 2, 3, 3 decimalplaces.Jahnke and Emde, Table XXV, first 5 roots for w = o, 3 for w = i, 2 forn = 2: 4 figures.Airey, 1. c. gives roots of some other equations involving Bessel's functionsconnected with the vibration of circular plates.Roots of: Jp(x)Yp(x) = Jp(kx)Yv{kx).Jahnke and Emde, Table XXVI, first 6 roots, 4 decimal places, forV = o, 1/2, I, 3/2, 2, 5/2: k = 1.2, 1.5, 2.0.Table XXVIII, first root, multiplied by (k - i) for ^ = i, 1.2, 1.5, 2-11,19, 39, «>: V same as above.Table XXIX, first 4 roots, multipUed by {k - i) for certain irrational valuesof k, and j^ = o, I. X. NUMERICAL SOLUTION OFDIFFERENTIAL EQUATIONSBy F. R. Moulton, Ph.D.,Professor of Astronomy, University of Chicago;Research Associate of the Carnegie Institution of Washington.INTRODUCTIONDifferential equations are usually first encountered in the final chapter ofa book on integral calculus. The methods which are there given for solvingthem are essentially the same as those employed in the calculus. Similar methodsare used in the first special work on the subject. That is, numerous types ofdifferential equations are given in which the variables can be separated bysuitable devices; little or nothing is said about the existence of solutions ofother types, or about methods of finding the solutions. The false impressionis often left that only exceptionally can differential equations be solved. What-ever satisfaction there may be in learning that some problems in geometry andphysics lead to standard forms of differential equations is more than counter-balanced by the discovery that most practical problems do not lead to suchforms.10.01 The point of view adopted here and the methods which are developedcan be best understood by considering first some simpler and better knownmathematical theories. SupposeI. F(x) = X" + aix"-i -1- -1- an-ix + an = ois a polynomial equation in x having real coefficients Oi, 02, . . . , On- If n isI, 2, 3, or 4 the values of :k which satisfy the equation can be expressed as explicitfunctions of the coefficients. If n is greater than 4, formulas for the solutioncan not in general be written down. Nevertheless, it is possible to prove that nsolutions exist and that at least one of them is real if n is odd. If the coefficientsare given numbers, there are straightforward, though somewhat laborious,methods of finding the solutions. That is, even though general formulas forthe solutions are not known, yet it is possible both to prove the existence of thesolutions and also to find them in any special numerical case.10.02 Consider as another illustration the definite integralI. /= ff(x)dx,where /(x") is continuous for a^x^ h. If F{x) is such a function that2. T--m, NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONSthen I = F{b) - F{a). But suppose no F(x) can be found satisfying (2). Itis nevertheless possible to prove that the integral / exists, and if the value of(x) is given for every value of x in thd interval a ^ x ^ b, it is possible to find thenumerical value of / with any desired degree of approximation. That is, it isnot necessary that the primitive of the integrand of a definite integral be knownin order to prove the existence of the integral, or even to find its value in anyparticular example.10.03 The facts are analogous in the case of differential equations. Thosehaving numerical coefficients and prescribed initial conditions can be solvedregardless of whether or not their variables can be separated. They need tosatisfy only mild conditions which are always fulfilled in physical problems.It is with a sense of relief that one finds he can solve, numerically, any particularproblem which can be expressed in terms of differential equations.10.04 This chapter will contain an account of a method of solving ordinarydifferential equations which is applicable to a broad class including all thosewhich arise in physical problems. A large amount of experience has shown thatthe method is very convenient in practice. It must be understood that there isfor it an underlying logical basis, involving refinements of modern analysis,which fully justifies the procedure. In other words, it can be proved that theprocess is capable of furnishing the solution with any desired degree of accuracy.The proofs of these facts belong to the domain of pure analysis and will not begiven here.10.10 Simpson's Method of Computing Definite Integrals. The method ofsolving differential equations which will be given later involves the computationof definite integrals by a special process which will be developed in this and thefollowing sections.Let t be the variable of inte-gration, and consider the definiteintegral1. F = ff(t)dt.This integral can be interpretedas the area between the /-axis andthe curve y = f{t) and boundedby the ordinates t = a and / = h,figure I.Let h = a, tn = b, yi = /(/,), anddivide the interval o ^ / ^ 6 up into11 equal parts, each of length h =(b - a) In. Then an approximate value of F is2. Fo = /^(j'i-hj2 + . . . + A'„).This is the sum of rectangles whose ordinates, figure i, are ji, yi,10.11 A more nearly exact value can be obtained for the first two intervals,for example, by putting a curve of the second degree through the three points Fig. 3'n- 222 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSyo, yi, yz, and finding the area between the /-axis and this curve and boundedby the ordinates /o and h. The equation of the curve is1. y = ao + ai{t - to) + 02(1 - /o)^where the coefficients Oo, Oi, and 02 are determined by the conditions that yshall equal yo, ji, and }'2 at / equal to to, /i and /a respectively; orf yo = ao,2. \ yy = Oo + a\{ti - to) + 02 (/i - /o)^[ 3'2 = flo + Ci(/2 - to) + 00(^2 - to)^.It follows from these equations and t2 - ti = ti - to = h thatOo = NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS •23 Ai>'2 = J2Al}'n = yn- Jn-i,These are the first differences of the values of the function y for successive valuesof /. All the successive intervals for t are supposed to be equal.10.21 In a similar way the second differences are defined byAiyz = Ai3'2 - Aiji,A2J3 = AO'3 - Ai>'2,Aij„ - Aij„_i,10.22 In a similar way third differences are defined byA3)'3 = Aojs - A2}'2,A3J4 = A2>'4 - A2>'3,A3>'n = Ao}'™ - A2A'„-1, and obviously the process can be repeated as many times as may be desired.10.23 The table of successive differences can be formed conveniently from thetabular values of the function and can be arranged in a table as follows: 224 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSnumbers in the various difference columns are zero. Now in such functionsas ordinarily occur in practice the numerical values of the differences, if theintervals are not too great, decrease with rapidity and run smoothly. If anerror is present, however, the differences of higher order become very irregular,10.25 As an illustration, consider the function y = sin t for t equal to io°,15°, The following table gives the function and its successive differ-ences, expressed in terms of units of the fourth decimal:^Table II t NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 225 will be affected by an error in the value of the function. The erroneous numbersin the last column are clearly the second, third, fourth, and fifth. The algebraicsum of these four numbers equals the sum of the four correct numbers, or -18.Their average is -4.5. Hence the central numbers are probably -5 and -4.Since the errors in these numbers are -36 and +36, it follows that € is probably+ 2. The errors in the second and fifth numbers are +€ and -e respectively.On making these corrections and working back to the first column, it is foundthat 7073 should be replaced by 7071. 10.30 Computation of Definite Integrals by Use of Difference Functions.Suppose the values of /(/) are known for t = tn-2, tn-i, tn, and /„+i. Supposeit is desired to find the integral = / m dt.The coefficients &o, ^i, h, and bs of the polynomial can be determined, as above,so that the function2. y = ^-0 + h{t - tn) + hit - t„Y + hit - tnYshall take the same values as/(/) for / = /„_2, ^»-i, t„, and /„+i.With this approximation to the function /(/), the integral becomes (sincetn+l - tn = h) 'tn+i + hit- tn) + hit - tnY + hit - /n)^] dtJf-tn+i 2 3 4 -^The coefficients ha, h, h, and bs will now be expressed in terms of y„+i, Aiy„+i,A2y„+i, and Asy^+i. It follows from (2) thaty„_2 = bo - 2bih + 4^^ - Sbsh^,yn-i = bo - bih + boh^ - bsh^,bo,[ yn+i = bo + bih +' boh- + bsh^.! ynThen it follows from the rules for determining the difference functions that . f Aiy„_a = bih - sb^Ji^ + 'jb.h',5. \ Aiy„ = bjt - boh'' + bji\{ Aiy„+i = bxh + bjt" + bzh\f Aay^ = 2bih'' - 6bzh\\ Azyn+i = 2b2h^.7. Asy^+i = 6bsh\ 2 26 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSIt follows from the last equations of these four sets of equations thatbo = yn+i - Ai>;„+i,Ihh = Ab2h^ = - A23'„+i, bih^ = - Asyn+u lyn+l - - A2>'„+1 Asjn+l, Therefore the integral (3) becomes9. In = h\ yn+l Aijn+i A^yn+l Asjn+l ......The coefficients of the higher order terms A4'v„4_i and A^yn+i are ^ and720 — respectively.4810.31 Obviously, if it were desired, the integral from /„_2 to /„_i, or over anyother part of this interval, could be computed by the same methods. For example,the integral from /„_i to /„ isrn-i= C'ki)dt, = }l\ yn+l - ^Aiy„+i + ^A2>'„+1 + ^AsVn+i +.....NUMERICAL ILLUSTRATIONS10.32 Consider first the application of Simpson's method. Suppose it is requiredto find I sin / dt cos / = 0.3327.On applying 10.12 with the numbers taken from Table I, it is found that1-0/i = —[.4226 + 2.0000 + 1. 1472 + 2.5712 + 1.4142 + 3.0640 + .8191],which becomes, on reducing 5° to radians,/i = 0.3327,agreeing to four places with the correct result.10.33 On applying 10.11 (4) and omitting alternate entries in Table II, it isfound that / = I sin t dt = — [.4226 + 2.2944 + .70713 = 0.1992,which is also correct to four places. These formulas could hardly be surpassedin ease and convenience of application. NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 22710.34 Now consider the application of 10.30 (9). As it stands it furnishes theintegral over the single interval /„ to /„+i. If it is desired to find the integralfrom /„ to tn+m, the formula for doing so is obviously the sum of m formulassuch as (9), the value of the subscript going from w + i to » + m + i, or/„, m = /ZH y.+l + + }'»+m+l j - \ (Ai}'„+1 + + AO'n+-+l) - ^ Uliyn+l + .... + A2j„4.™+1 j - ^ ( A3}'n+1 +.... + A33'„+^+ij + . . . •On applying this formula to the numbers of Table I, it is found that J'* 55°sin / dt = 5°[(.5ooo + .5736 + .6428 + .7071 + .7660 + .8191)25° - - (.0774 + .0736 + .0692 + .0643 + .0589 + .0531)+ — (.0032 + .0038 + .0044 + .0049 + .0054 + .0058) -\ (.0006 + .0006 + .0006 + .0005 + .0005 + .0004)3 = 0.3327,agreeing to four places with the exact value. When a table of differences is athand covering the desired range this method involves the simplest numericaloperations. It must be noted, however, that some of the required differencesnecessitate a knowledge of the value of the function for earlier values of theargument than the lower limit of the integral. 10.40 Reduced Form of the Differential Equations. Differential equationswhich arise from physical problems usually involve second derivatives. Forexample, the differential equation satisfied by the motion of a vibrating tuningfork has the form d~xdF = -'"'where ^ is a constant depending on the tuning fork.10.41 The differential equations for the motion of a body subject to gravityand a retardation which is proportional to its velocity ared~x _ dxdF~ '"^dt'dry dy^It^^-'dl-^^where c is a constant depending on the resisting medium and the mass and shapeof the body, while g is the acceleration of gravity. 228 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS10.42 The differential equations for the motion of a body moving subject tothe law of gravitation are df ~ r^'^_ _]e^y_dt' ~ r^'df ' r^' r2 = x" + f + s2.10.43 These examples illustrate sufficiently the types of differential equationswhich arise in practical problems. The number of the equations depends onthe problem and may be small or great. In the problem of three bodies thereare nine equations. The equations are usually not independent as is illustratedin 10.42, where each equation involves all three variables x, y, and z through r.On the other hand, equations 10.41 are mutually independent for the first doesnot involve y or its derivatives and the second does not involve x or its deriva-tives. The right members may involve x, y, and z as is the case in 10.42, orthey may involve the first derivatives, as is the case in 10.41, or they mayinvolve both the coordinates and their first derivatives. In some problemsthey also involve the independent variable t.10.44 Hence physical problems usually lead to differential equations which areincluded in the form df J., dx d^^^''^y'dt'dt''d'ydf g[x,y: dx dydi'di .),where / and g are functions of the indicated arguments. Of course, the numberof equations may be greater than two.10.45 If we let NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 22910.46 If we let x = xi, x' = xi, y = xz, y' = Xi, equations 10.45 areincluded in the form [ dxi , I -^ =fAXi,Xi,. . . ,Xn,t), dXn r r ,NThis is the final standard form to which it will be supposed the differentialequations are reduced. 10.50 Definition of a Solution of Differential Equations. For simplicity inwriting, suppose the differential equations are two in number and write them inthe form where / and g are knovv^n functions of their arguments. Suppose x = a, y = hat / = o. Then is the solution of (i) satisfying these initial conditions if <\) and 1/' aresuch functions that (o) = a, the last two equations being satisfied for all o ^ i^T, where T is a positive con-stant, the largest value of t for which the solution is determined. It is not neces-sary that <\) and -^ be given by any formulas — it is sufficient that they havethe properties defined by (3). Solutions always exist, though it will not beproved here, iff and g are continuous functions of t and have derivatives with respectto both X and y.10.51 Geometrical Interpretation of a Solution of Differential Equations.Geometrical interpretations of definite integrals have been of great value notonly in leading to an understanding of their real meaning but also in suggesting Fig. 2 230 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSpractical means of obtaining their numerical values. The same things are truein the case of differential equations.For simphcity in the geometrical representation, consider a single equation where x = a Sit t = o. Suppose the solution is2. X = = a + Jj((f), 4/, t) dt, ^ = b + £g{(j>, xP, t) dt.The difficulty arises from the fact that and 1/' are not known in advance andthe integrals on the right can not be formed. Since <^ and xp are the solutionvalues of x and y, we may replace them by the latter in order to preserve theoriginal notation, and we have NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 23 1X = a + £f{x, y, t) dt, \ y = b + jj{x,y,t)dLIf X and y do not change rapidly in numerical value, then/(x, y, t) and g{x, y, t)will not in general change rapidly, and a first approximation to the values of xand y satisfying equations (2) isf xi = a + fj{a, b, t) dt,Ui = * + \gia, b, t) dt,at least for values of / near zero. Since a and b are constants, the integrands in(3) are known and the integrals can be computed. If the primitives can not befound the integrals can be computed by the methods of 10.1 or 10.3.After a first approximation has been found a second approximation is given by 4. [x2 = a + / /(x„ y., /) dt,\ Jo\yt = b + j g(xi,yi,t) dt.The integrands are again known functions of / because Xi and yi were determinedas functions of / by equations (3). Consequently X2 and y2 can be computed.The process can evidently be repeated as many times as is desired. The nthapproximation is = a+ J f(Xn-l, yn-1, t) dt.yn = b + J g(Xn-l, yn-1, t) dt^There is no difficulty in carrying out the process, but the question arises whetherit converges to the solution. The answer, first established by Picard, is that,as n increases, Xn and y„ tend toward the solution for all values of / for which allthe approximations belong to those values of x, y, and / for which / and g havethe properties of continuity with respect to t and differentiability with respectto X and y. If, for example, / = —— and the value of Xn tends towards zerofor t = T, then the solution can not be extended beyond / = T.It is found in practice that the longer the interval over which the integrationis extended in the successive approximations, the greater the number of approxi-mations which must be made in order to obtain a given degree of accuracy. Infact, it is preferable to take first a relatively short interval and to find the solutionover this interval with the required accuracy, and then to continue from the endvalues of this interval over a new interval. This is what is done in actual work.The details of the most convenient methods of doing it will be explained in thesucceeding sections. 232 MATHEMATICAL FORilULiE AND ELLIPTIC FUNCTIONS 10.7 The Step-by-Step Construction of the Solution. Suppose the differentialequations are I J=/fey,o, with the initial conditions x = o, j = 6 at / = o. It is more difficult to start asolution than it is to continue one after the first few steps have been made. There-fore, it will be supposed in this section that the solution is well under way, andit will be shown how to continue it. Then the method of starting a solution willbe explained in the next section, and the whole process will be illustratednumerically in the following one.Suppose the values of x and y have been found for / = /i, ^2, . • . • , h. Letthem be respectively Xi, yi\ x^, y^; . . .; x„, y„, care being taken not to confusethe subscripts with those used in section 10.6 in a different sense. Suppose theintervals to — /,, ti — to, . . . , /„ — /„_i are all equal to h and that it is. desiredto find the values of x and y at /„+i, where /„+i - tn = h.It follows from this notation and equations (2) of 10.6 that the desiredquantities are Xn+i = Xn+ j f{x,y, t) dt,Mn+Tyn+1 = yn+ I g {x, y, t) dt.The values of x and y in the integrands are of course unknown. They can befound by successive approximations, and if the interval is short, as is supposed,the necessary approximations will be few in number.A fortunate circumstance makes it possible to reduce the number of approxi-mations. The values of x and y are known at / = /„, tn-i, tn-%, . • . From thesevalues it is possible to determine in advance, by extrapolation, very close approxi-mations to X and y ior t = tn+i- The corresponding values of / and g can becomputed because these functions are given in terms of x, y, and /. They arealso given for t = tn, tn-i, Consequently, curves for / and g agreeingwith their values at t = tn+i, tn, tn-i, .... can be constructed and the integrals(2) can be computed by the methods of 10.1 and 10.3.The method of extrapolating values of .t„+i and y„+i must be given. Sincethe method is the same for both, consider only the former. Since, by hypothesis,X is known for t = tn, t„-i, tn-2, .... the values of Xn, Aia:„, ^^Xn, and^zXn are known. If the interval h is not too large the value of i^zXn+\ is verynearly equal to AsX™. As an approximation i^zXn+\ rnay be taken equal to Aj-Tn,or perhaps a closer value may be determined from the way the third differences NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 233Aa.Tn-s, AsXn-i, AzXn-i, and AzXn vary. For example, in Table II it is easy to seethat A3 sin 75° is almost certainly -3. It follows from 10.20, 1, 2 that( A-iXn+i = As-T^+i + AoXn,3. I AiXn+1 = Aa.T^+i + Ai.Tn,[ Xn+1 = AiXn-fi + Xn-After the adopted value of A3X,t+i has been written in its column the successiveentries to the left can be written down by simple additions to the respec-tive numbers on the line of /„. For .example, it is found from Table II thatAo sin 75° = -72, Ai sin 75° = 262, sin 75° = 9659. This is, indeed, the correctvalue of sin 75° to four places.Now having extrapolated approximate values of Xry^-i and y„+i it remains tocompute / and g for x = Xn+i, y = Jn+i, t = /„+i. The next step is to pass curvesthrough the values of/ and g for / = ^„+i, tn, tn-i, .... and to compute the inte-grals (2). This is the precise problem that was solved in 10.30, the only differencebeing that in that section the integrand was designated by y. On applyingequation 10.30 (9) to the computation of the integrals (2), the latter give 234 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS10.8 The Start of the Construction of the Solution. Suppose the differentialequations are again with the initial conditions x = a,y = bSitt = o. Only the initial values of x andy are known. But it follows from (i) that the rates of change of x and ^^ at / = oare/(o, b, o) and g(a,b , o) respectively. Consequently, first approximations tovalues of x and y a,t t = h ^ h are/ .Ti(i) = a + hj{a, b, o),^' \y,^'^ = b + hg(a,b,o).Now it follows from (i) that the rates of change of x and y at x = Xi, y = yi,t = ti are approximately /(x/^), yi^^\ /i) and ^(xi^'), yi^\ ti). These rates will bedifferent from those at the beginning, and the average rates of change for thefirst interval will be nearly the average of the rates at the beginning and at theend of the interval. Therefore closer approximations than those given in (2) tothe values of x and y SLt t = ti are3. / ^1^'^ = « + hh [ fia, b, o) + fix,^'), j,(i), /O],1 j,(^-) ^b + ^h [g(a, b, o) + g(x/'), yS'\ /:)].The process could be repeated on the first interval, but it is not advisable whenthe interval is taken as short as it should be.The rates of change at the beginning of the second interval are approximatelyf{xi^^\ yi^^\ ti) and g{xi^^\ yS^\ U) respectively. Consequently, first approxima-tions to the values of x and 3^ at / = ti, where h - h = h, are4. / X^'l) = XP + /7/(.Vi(2), y,(~^), U),\yS'^ = yP + hg{x,^''\yS'\h).With these values of x and }' approximate values of /2 and gi are computed. Since/o, go; /i, gi are known, it follows that A1/2, ^ig-,; A2/2, and A2g2 are also known.Hence equations (4) of 10.7, for w + i = 2, can be used, with the exception ofthe last terms in the right members, for the computation of X2 and yi.At this stage of work Xo = a, yo = b\ Xi, yi; x^, y-i are known, the first pairexactly and the last two pairs with considerable approximation. After ji and gthave been computed, Xi and yi can be corrected by 10.31 for « = i. Then ap-proximate values of Xz and yz can be extrapolated by the method ex-plained inthe preceding section, after which approximate values of /s and gz can be com-puted. With these values and the corresponding difference functions, x-i and y^can be corrected by using 10.31. Then after correcting all the correspondingdifferences of all the functions, the solution is fully started and proceeds by themethod given in the preceding section.10.9 Numerical Illustration. In this section a numerical problem will be treatedwhich will illustrate both the steps which must be taken and also the method of NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 235arranging the work. A convenient arrangement of the computation which pre-serves a complete record of all the numerical work is very important.Suppose the differential equation is I^ = - (i + k2).v + 2/cV,1 dxa- = o, -7- = I at / = o.l^ atThe problem of the motion of a simple pendulum takes this form when expressedin suitable variables. This problem is chosen here because it has an actual physi-cal interpretation, because it can be integrated otherwise so as to express / interms of x, and because it will illustrate sufficiently the processes which havebeen explained.Equation (i) will first be integrated so as to express / in terms of x.On multiplying both sides of (i) by 2 — and integrating, it is found that theintegral which satisfies the initial conditions is(|Y.(,_,,.^)(,_.3,,),On separating the variables this equation givess:K V (i - .1-2) (i - /cV)Suppose V? <\ and that the upper Umit x does not exceed unity. ThenI I ^ c;4. / = I + - K-.V- + f K^X^ + \ K^X^ + . . . .V I - K-x- 2 8 16where the right member is a converging series. On substituting (4) into (3) andintegrating, it is found that5. ^ = sin 1 .r + i[-.vVi - x^ + sin-i xY + f[-xVi - x^ - \x{\ - .t^)?+ f.vVi - y? + f sin-i x\k^ ^ ].When X = I this integral becomes Equation (5) gives t for any value of x between - 1 and 4-1. But the problemis to determine x in terms of /. Of course, if a table is constructed giving / formany values of .r, it may be used inversely to obtain the value of x correspondingto any value of /. The labor involved is very great. When tC- is given numericallyit is simpler to compute the integral (3) by the method of 10.1 or 10.3.In mathematical terms, / is an elliptical integral of x of the first kind, and theinverse function, that is, x as a function of /, is the sine-amplitude function, whichhas the real period /\,T. 236 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSSuppose K" = NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 237First Trial j-Table / 238 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSafter which it is found that gi = -.i486, Aigi = -.i486. Now the first trial }'-tablecan be corrected by using the value of ji given in (14). The result is: NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 239first steps are very simple and can be carried out in practice in a few minutes ifthe chosen time-interval is not too great.The problem now reduces to simple routine. There are an x-table, a y-table(which in this problem serves also as an /-table), a g-table, and a schedule forcomputing g. It is advisable to use large sheets so that all the computationsexcept the schedule for computing g can be kept side by side on the same sheet.The process consists of six steps: (i) Extrapolate a value of g„+i and itsdifferences in the g-table; (2) compute yn+i by the second equation of 10.7 (4);(3) enter the result in the y-table and write down the differences; (4) use theseresults to compute .t„+i by the first equation of 10.7 (4) ; (5) with this value ofXn+i compute g„+i by the g-computation schedule; and (6) correct the extrapolatedvalue of g„+i in the g-table.Usually the correction to g„+i will not be great enough to require a sensiblecorrection to y„+i. But if a correction is required, it should, of course, be made.It follows from the integration formulas 10.7 (4) and the way that the differencefunctions are formed that an error e in g„+i produces the error ^he in yn+i, andthe corresponding error in Xn+i is — h^e.04 It is never advisable to use so largea value of h that the error in Xn+i is appreciable. On the other hand, if the differ-ences in the g-table and the y-table become so small that the second differencesare insensible the interval may be doubled.The following tables show the results of the computations in this problemreduced from five to four places. Final .r-Table / 240 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONS NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS 241 Final ?-Table / 242 MATHEMATICAL FORMULA AND ELLIPTICAL FUNCTIONSAs has been remarked, large sheets should be used so that the x, y, and g-tablesean be put side by side on one sheet. Then the /-column need be written but oncefor these three tables. The g-schedule, which is of a different type, should be ona separate sheet.The differential equation (i) has an integral which becomes for k^ = -2 ' , dxand ^ = y. 7 I21. y + -.^2 a-* = I,2 4and which may be used to check the computation because it must be satisfied atevery step. It is found on trial that (21) is satisfied to within one unit in thefourth place by the results given in the foregoing tables for every value of /.The value of / for which :*; = i and y = o is given by (6) . When /c^ = i it isfound that T = 1.8541. It is found from the final .r-table by interpolation basedon first and second differences that x rises to its maximum unity for almost exactlythis value of /; and, similarly, that y vanishes for this value of /. XI ELLIPTIC FUNCTIONSBy Sir George Greenhill, F. R. S. INTRODUCTION TO THE TABLES OF ELLIPTICFUNCTIONSBy Sir George GreenhillIn the integral calculus, / —/rr' and more generally, / "^ ^_ dx«/ V^ J P + QVX 'where M, N, P, Q are rational algebraical functions of x, can always be expressedby the elementary functions of analysis, the algebraical, circular, logarithmic orhyperbolic, so long as the degree of X does not exceed the second. But whenX is of the third or fourth degree, new functions are required, called ellipticfunctions, because encountered first in the attempt at the rectification of anellipse by means of an integral.To express an elliptic integral numerically, when required in an actualquestion of geometry, mechanics, or physics and electricity, the integral mustbe normalised to a standard form invented by Legendre before the Tables canbe employed; and these Tables of the Elliptic Functions have been calculatedas an extension of the usual tables of the logarithmic and circular functions oftrigonometry. The reduction to a standard form of any assigned elliptic integralthat arises is carried out in the procedure described in detail in a treatise on theelliptic functions.11.1. Legendre's Standard Elliptic Integral of the First Kind (E. I. I) isdxa-2)(i - /cV)defining 4> as the amplitude of u, to the modulus k, with the notation, = am wX = sin (^ = sm am uabbreviated by Gudermann to,X = sn wcos <^ = en wA = vCi - /<:' sin^ ^) = A am w = dn w,and sn w, en w, dn u are the three elliptic functions. Their differentiations are,d'making sn ii^ = Ien K = odn A' = k'.k' is the comodulus to k, k^ + k"^ = i, and the coperiod, K\ is,d4)Jo V(i - k'^ sin2 0)11.12. sn- w + cn^ M = Icn^ M + K^ sn^ w = Idn^ u - K^ en- m = k'^.sn o = o, en o = dn, 0=1.sn K = I, en A' = o, dn A = k' .11.13. Legendre has calculated for every degree of 6, the modular angle,K = sin 6, the value of /^0 for every degree in the quadrant of the amplitude 0,and tabulated them in his Table IX, Fonctions elliptiques, t. II, 90 X 90 = 8100entries.But in this new arrangement of the Table, we take u = F4> as the independentvariable of equal steps, and divide it into 90 degrees of a quadrant A, puttingr°u = eK = —5 A, r° = 90°^.90As in the ordinary trigonometrical tables, the degrees of r run down the left ofthe page from 0° to 45°, and rise up again on the right from 45° to 90°. Thencolumns II, III, X, XI are the equivalent of Legendre's Table of F4> and (j),but rearranged so that F^ proceeds by equal increments 1° in r°, and the incre-ments in 4> are unequal, whereas Legendre took equal increments of givingunequal increments in w = F(^.The reason of this rearrangement was the great advance made in ellipticfunction theory when Abel pointed out that F(/) was of the nature of an inversefunction, as it would be in a degenerate circular integral with zero modularangle. On Abel's recommendation, the notation is reversed, and <^ is to be INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 247considered a function of u, denoted already by = am u, instead of lookingat u, in Legendre's manner, as a function, F4>, of cj). Jacobi adopted the ideain his Fundamenta nova, and employs the elliptic functionssin 4> = sin am u, cos 4> = cos am u, A(f) = A am u,single-valued, uniform, periodic functions of the argument u, with (quarter)period K, as cf) grows from o to |7r. Gudermann abbreviated this notation tothe one employed usually today.11.2. The E. I. I is encountered in its simplest form, not as the elliptic arc,but in the expression of the time in the pendulum motion of finite oscillation,unrestricted to the small invisible motion of elementary treatment.The compound pendulum, as of a clock, is replaced by its two equivalentparticles, one at in the centre of suspension, and the other at the centre ofoscillation, P; the particles are adjusted so as to have the same total weight asthe pendulum, the same centre of gravity at G, and the same moment of inertiaabout G or 0; the two particles, if rigidly connected, are then the kinetic equiva-lent of the compound pendulum and move in the same way in the same field offorce (Maxwell, Matter and Motion, CXXI).Putting OP = I, called the simple equivalent pendulum length, and P startingfrom rest at 5, in Figure i, the parti-cle P will move in the circular arcBAB'sLS if slidingdown a smooth curve ; and P will acquire the same velocityas if it fell vertically KP = ND; thisis all the dynamical theory required.(velocity of P)- = 2g-KP,(velocity of Ny= 2g-ND- sin^^OP2g-ND- NP^_g_2OP'- ~ P ND-NA-NE,and with AD = h, AN = y, ND= h- y, AE = 2I, NE^ 2I- y, where Y is a cubic in y. Then t is givenby an elliptic integral of the form Fig. i/dv . . . . —^- This integral is normalised to Legendre's standard form of hisE. I. I by putting y = h sin^ (/>, making AOQ = (f), h - y = h cos- (j),2I - y = 2I {1 - Hr sin- 0), 2I AEK is called the modulus, AEB the modular angle which Legendre denotedby 0; V(i - K^sm'^(f)) he denoted by Acp. 248 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSWith g = M, and reckoning the time / from A, this makes in Legendre's notation. Then the angle 4> is called the amplitude of nt, to bedenoted am nt, the particle P starting up from A at time / = o; and with u = 7it,AP AQ „ AN''''' = AB = AD ''''' = ADDQ „ PKcnu = j^ cn^« = ^ , EP , ^ NEdnu = -=-r dn^ w = -r^EA AEVelocity of P = n-AB-cn u = y/BP-PB', with an oscillation beat of T secondsin M = eK, e = 2t/T.11.21. The numerical values of sn, en, dn, tn (m, k) are taken from a tableto modulus K = sin (modular angle, d) by means of the functions Dr, Ar, Br,Cr, in columns V, VI, VII, VIII, by the quotients.V/c'snei^ INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS !49ds = Va^ COS- 4) -\- b~ sin- (^ = aA{4>, k),d(l>to the modulus k, the eccentricity of the eUipse.Then s = a Ecf), whereJo A0 • dcf) is denoted by E(f>in Legendre's notation of his standard E. I. II;it is tabulated in his Table IX alongside of F4>for every degree of the modular angle 6, and toevery degree in the quadrant of the amplitude <^.But it is not possible to make the inversionand express as a single-valued function of £. Fig.11.31. The E. I. II, E(f), arises also in the expression of the time, /, in the oscil-lation of a particle, P, on the arc of a parabola, as F4> was required on the arcof a circle. Starting from B along the parabolaBAB', Figure 3, and with AO = //, OB = b,BOQ = (j), AN = y = h cos^ ,NP = x = b cos(^ and with OS = 2h = b tan a, OA' = SB= b sec a, the parabola cutting the horizontalat B at an angle a, the modular angle, BRA'B'is a semi-ellipse, with focus at S, and eccen-tricity K = sin a.V, Wio / //—\ /\ y p // (Velocity of Py + 'l Fig. 3 (&2 cos^ ^ -(- 4F sin^ = a~, Vt^s,d

is replaced by the columns IV, IX, of E{r) andG{r) = £(90 - r), defined, in Jacobi's notation, byE{r) = zn eK = E4> - eEG(r) = zn (i — e)K, r = Qoe.This is the periodic part of E(() after the secular term eE = —u has been setaside, E denoting the complete E. I. II,£ = £i7r = /^"A(/)-(/0.The function zn u, or Zu in Jacobi's notation, or £(r) in our notation, iscalculated from the series, r^ r, TT '^ sin 2nir 27r ^ / ™ ,-!„,, .i,„ , \ •Er = Zu = —y^ p = ~y^ {q^^ + (f"' + q^'" + . . . .) sm 2mr.r^ sinh nnr -^ ;^~This completes the explanation of the twelve columns of the tables.11.4. The Double Periodicity of the Elliptic Functions.This can be visualised in pendulum motion if gravity is supposed reversedsuddenly at B (Figure i) the end of a swing; as if by the addition of a weightto bring the centre of gravity above O, or by the movement of a weight, as in themetronome. The point P then oscillates on the arc BEB' , and beats the ellipticfunction to the complementary modulus k' , as if in imaginary time, to imaginaryargument nti = JK'i: and it reaches P' on AX produced, where tan AEP'= tan AEB- en {nt'i, k), or tan EAP' = tan EAB- en {nt', k'); or with nt' = f,DR' = DB-zn {iv, k'), DR = DB- en {v, k'), 'with DR-DR' = DB\ EP' crossingDB in R'. en {iv, k) en {v, INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 25 111.42. Coamplitude Formulas, with v = ± K,sn (A - w) = :5 = sn (A + u)dnu , ^, . k' snu ,^^ . k' sn wen (A - u) = —, en (A + w) = -.^ ' dnw ^ ^ dnwdn (A - u) = -5 = dn (A + u)^ dn wtn {K - u) = -TT— tn (A + w) = 7- — ' K tnu K tnu11.43. Legendre's Addition Formula for his E. I. II,Ecj) = fA(f)-d(f) = fdv^u-du, (p = J'dnu-du = am w.E(j) + Exp — E) = /c^ sn w sn d sn {v + w),the secular part cancelling.Another form of the Addition Theorem for Legendre's E. I. II, ^ ^^ ^ , - 2 K^ sin i/' cos i/' Ai^ sin^ ^ ^Ed - Ed - lExp = , . „ J • 2 >—^, = am (7; - w)^ 1 - K^ sm^ ^ sm^ Yor, in Jacobi's notation, . . , , — 2 K^ sn ?^ en i> dn I) sn^ wzn iv + u) + zn iv — u) — 2 zn i; = 5— 5 5 — •^ ^ ^ ^ I - /c^ sn^ w sn^ z;11.5. The Elliptic Integral of the Third Kind (E. I. Ill) is given by the nextintegration with respect to u, and introduces Jacobi's Theta Function, Qu,defined by, d log Qu _f = Zw = zn Mdu -Qo = ^-^"P- X™ '''^'''Integrating then with respect to u, , „ . . , ^ . . T- 2 K^ sn 2; en 2) dn 7; sn2 Mlog B (v + u) - log i3 (v - u) - 2U znv = I 5— ^ 5—— aw,^ ^ ^ ^ ^ ^ Jo I - K^ sn^ w sn^ yand this integral is Jacobi's standard form of the E. I. Ill, and is denoted by — 2n (m, t)) ; thus,TT . , Ck"^ snv cnv dnvsn^u J ,1, Q (v - u)11 (w, V) = I , „ —s du = u znv + ^ log q . ,—r •^ ' ' J I - /c2 sn2 M sn2 1) ^ ^ O {v + u) [ byVk sn V,Jacobi's Eta Function, Kv, is definedmGvand then d log Ht) en 7; dn y + zn z;, denoted by zs v;dv snv 252 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSSO that en t; dn^; ,du J''" "' en t) an ?; ^t / \u h 11 (m, V)K" sn' u sn' y I , Q{v -u) 2 * (s + a)/.This gives Legendre's standard E. I. Ill,M d^I + n sin^ (j) A(/)where we put n = - k^ sri^ v = - k^ sin^ xf/,I \C-\ , . eos^ yhl\^\h cn^ z; dn^ vV w / sin^ yp sn^ z) 'the normalising multiplier, M.The E. I. Ill arises in the dynamics of the gyroscope, top, spherical pendulum,and in Poinsot's herpolhode. It can be visualized in the soHd angle of a slantcone, or in the perimeter of the reciprocal cone, a sphero-conic, or in the mag-netic potential of the circular base.11.51. We arrive here at the definitions of the functions in the tables. Jacobi's9m and Hm are normalised by the divisors Go and Hi^L, and with r = gee,D{r) denotes "o^; ^W denotes -^jr—while B{f) = ^(90 - r), Cij) = D{go - r), and B(o) = A(go) = D(o) = C(go) = 1, C(o)=D(9o) = -^-VkThen in the former definitions,A(r) A (go) r-, ^7^7-r = 7w—r sn M = V K sn eKD{r) D{go)B(r) B(o) ^rvT = T^r^ en w = en eKDir) Dip)C{r) C(o) , dn^ii:D{r) Dio) Vk'Then, with u = eK, v = fK, r = 90^, 5 = 90/, „„..!, D (s - r) zn/iC . £ (s), zn (i - /) A' - £ (90 -s) =G{s). INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 253The Jacobian multiplication relations of his theta functions can then berewritten D(r + s)D{r - s) = D-rDh - tan^ dAhA^s,A(r + s)A{r - s) = AhDh - DhAh,B{r + s)B{r - s) = BhBh - AhAh.But unfortunately for the physical apphcations the number 5 proves usuallyto be imaginary or complex, and Jacobi's expression is useless; Legendre callsthis the circular form of the E. I. Ill, the logarithmic or hyperboHc form corre-sponding to real 5. However, the complete E. I. Ill between the limits o <0 <|7r,or o s>si/•°°-i./c, _ Co d<:eK Js -y/S V5-53 \ S - S3 y S - S3 X j^ rWsi - S3 ds s - Si Si - So si- S2-S - S3(i - e)K = I ;= = sn iV/ = en ^\ = dn ^SJJsi \/S ^ S - S2 y S - S2 ^ Si- S3-S - S2indicating the substitutions, = sin^ d> = sn^ eK. ^ = sin^ "d/ = sn^ (i - e)K.S - S3 s - S2In the next interval S is negative, and the comodulus k' is required. ^1 >S>S2 ' J V- S ^ Si- S2 y Si- S2 >^ Si- S3 . . , ( \/Sl - S3 ds Sx - S3-S - So _,. A2 - S3- Si - S{i-f)K' = I ; — = sn ^y = en ^V/Js, V -S ^Si-S2-S-S3 V 51-52-5-^1y S - S3S is positive again in the next interval, and the modulus is k.52 >5 >53 r M- r^'Vsi-Ssds _n Ai - 53-52 - 5 _,. Ai - 52-5 - 53(i - e)K = I 1= = sn^V = cn^V/Js -^S V 52 -53-51-5 » 52 -53-51-5 r'V5i - 53 ds S - S3 _,. /52 - 5 Ai . = I = = sn iV/ = en V = dn \ -Jh VS V 52 - 53 V 52 - 53 > 5ieK .indicating the substitutions, dn- INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 255S is negative again in the last interval, and the modulus k'.S3>S> - CO(I _ ^ rWs^-jsds ^ ^^.Js^-^ ^ en-V'-^^^ = dnV^--^-"--^Js V-S yS2-S \S2-S \si-S3-S2-SJK' = r ^^5^i£ = sn-V"^^ = cn-v/^Hi = dn"V^^J-00 V- 5 y si-s y si- s v^i-^11.8. For the notation of the E. I. II and the various reductions, take thetreatment given in the Trans. Am. Math. Soc, 1907, vol. 8, p. 450. The JacobianZeta Function and the Er, Gr of the Tables, are defined by the standard integralr ^'~^ -^ = / "^Ac/) • J0 = £(/) = r dn2 (eA') • d{eK) = EsimeK = eH + zn eK,Jh -ySi — S3 -vS *Jo Joor, l^.a-^ da ^ Pf^^^ (fK').d{fK') = E am /A' = JH' + zn/A", where zn is Jacobi's Zeta Function, and i7, W the complete E. I. II to modulus/c, k:', defined by, TVH = £^^((f>, k) d(}> = j^'dn^ {eK)-d{eK)H' = /Ja((/), k') d = J;dn=^ UK')-d{fK').The function zn u is derived by logarithmic differentiation of 9w, zn w = 256 MATHEMATICAL FORMULA AND ELLIPTIC FUNCTIONSn ' - '^ al = r^-^^ y^^' ^5 = - d - e)H + zsckJs Vsi -SsVS J^ ^- ^3 VS fj - '' A = f'A^fl Vs^ ds={i- e){K -H)+ zseKJ Vsi-ssVS -^ ^ - ^3 VSthe integrals being c° at the upper Umit, s = oo , or at the lower limit, s = S3where e = o and zs eK = 00 . So also,f'^'^s - S2 Vsi - 53 , _ P' '^ Si- s ds^ ^eH + zn eKJs, 5, ^ - ^3 V5 *^''' ' Vsi - ss VS (i - e)H -zn eKJ ^ - ^3 V5 "^ J V^^^Tz VS (i - e){H - k'^K) - zn eKCsi - 53 V^i - 53 ^^ ^ r s - ss ds^ ^eiK - H) - zn eA' Similarly, for the variable a in the regionsSi><7>S2>S3>(T> - °°2 negative, and sn^fK = or zn eA p- '^ si- (T da ^ C" ^3 sj^J' \/5i - 53 V - S/-^^-0-v^ r-A^^-i^ = -(i-/)F'+zs/A'J5i-(7V-S -'\/5i-53V-Sthese last three integrals being infinite at the upper limit, a- = 5i, or lower limit(7 = _ CO, where / = o, zs JK' = <» . Putting e = I or / = I any of these forms will give the complete E. I. II,noticing that zn A' and zs K' are zero. INTRODUCTION TO THE TABLES OF ELLIPTIC FUNCTIONS 257 '11.9. In dealing practically with an E. I. Ill it is advisable to study it firsiin the algebraical form of Weierstrass,A{s-aWSwhere S = 4-5 — si-s - s^-s — S3, S the same function of a, and begin by ex-amining the sequence of the quantities s, a, Si, S2, s^Then in the region s>Si>S2>cr>S3,put —5— , a - S3 = {s2 - S3) srv V, k- = »sn^ u' ' ' s^ - S-i ^1 - ^3 / 2 •> " ^ V ^1 - ssdss — a = —5— ( I - K'^ sn- u sn- v), = = du,sn-u ^sV2 = V^i — -^3 (.^2 — -^3) sn z) en i) dn v, making/iVS ds Z' /c^ sn ?; en i; dn t) sn^ z* , tt/ n~ —= = / 5 — o o du = n(w, v).s - (T ^ys J I - K- sn- M sn- VBut in the region, cr >5i >S2>s>S3,( so ^1 - ^3 I /v / vsCnTJdnvS - S3= \S2- S3) Sn'^ «, G - S3 = —5— 5 -yJZj = {Sy - S3)- 1 'sn^ z; 2 '&x^ VS\- S3 (i - K- sn-M sn^ v),sn- Vmaking, en 7; dn I) _,/— 7 ^ —~~ du ,WS ds r snz) ^ ^, . Qx^vAwvjXj^ —= = / ^— ^ TT = 111 = lUw, v) -{ ud - s y/S J I — /< sn- w sn- v sn z;In a dynamical apphcation the sequence is usuallyS>Si>a>S2>S>S3or S>Si >S2>S>S3>(T,making 2 negative, and the E. I. Ill is then called circular; the parameter vis then imaginary, and the expression by the Theta function, is illusory.The complete E. I. Ill, however, was shown by Legendre to be tractableand falls into four classes, lettered (V) (w'), P- 138, (i'), (k'), pp. 133, 134 (Fonc-tions elliptiques, I).Si>(7>S2 Si- (Tsn2 JK' cn2 fK' Si - S2a - S2Sy - S2a - S3Sl - S3 258A. B. C. D. MATHEMATICAL FORMLU^ AND ELLIPTIC FUNCTIONS , "-Zr^^ = A{fK') = irCi - /) - K znJK'VSA +B ^hir.S3>(T> - CO sn2 JK' = dnVA^' = 5i - ^3si - aS3- 5>5i v-i: ds0- V5 C(/A^') =KzsfK' -lTv{i-f) ''^'^'J^'t^ 7k = ^^^^^'^ = ^^^^^^' + ^"^D-C = ^7r. TABLES OF ELLIPTIC FUNCTIONSBy Col. R. L. Hippisley 26o ELLIPTIC FUNCTIONK = 1.5737921309, K' = 3.831742000, E = 1.5678090740, E' = 1.012663506, r TABLE = 5°(7=0.000476569916867, 6 0=0.9990468602, H(K) = 0.2955029021 26]B(r) 262 ELLIPTIC FUNCTIONK = 1.5828428043, K' = 3.153385252, E = 1.5588871966, E' = 1.040114396, r TABLE d = 10°q = 0.00191359459017, 6 = 0.9961728108, HK = 0.418305976553 263B(r) 264K = 1 . 5981420021, K' ELLIPTIC FUNCTIONKV3 = 2 . 7680631454, E = 1 . 5441504939, E' = 1 . 076405113, r TABLE d = 15°g = . 004333420509983, 6 = 0. 9913331597, HK = . 5131518035 265B(r) 266 ELLIPTIC FUNCTIOK = 1 . 6200258991, K' = 2 . 5045500790, E = 1 5237992053, E' = 1 , 118377738 r TABLE 9 = 20°g =0 007774680416442, 6 = 0. 9844506465, 2675939185400B(r) 268 K = 1.6489952185, K' ELLIPTIC FUNCTION3087867982, E = 1.4981149284, E' = 1.1638279645, r TABLE e = 25°q = 0. 012294560527181, 6 0=0. 975410924642, HK = 0. 666076159327 269B(r) 270 ELLIPTIC FUNCTIONK = 1. 6857503548, K' = 2 . 1565156475, E = 1. 4674622093 E' = 1. 211056028, r TABLE e = 30°q =0.017972387008967, 2716 0=0. 9640554346, HK = 0. 7325237222B(r) 272 ELLIPTIC FUNCTIONK = 1.7312451757, K' = 2.0347153122, E = 1.4322909693, £' = 1.2586796248, ro TABLE 9 = 35°g =0. 024915062523981, 6 = 0. 9501706456, HK = . 7950876364 273B(r) 274 K = 1. 7867691349, K' = 1. 9355810960, ELLIPTIC FUNCTIONE = 1. 3931402485, E' = 1. 3055390943, ro TABLE d = 40°q = 0.033265256695577, = 0. 9334719356, HK = 0. 8550825245 275B(r) 270 ELLIPTIC FUNCTIONK = K' = 1. 8540746773, E = E' = 1. 3506438810, r TABLEq = Q-'^ = 45°0.04321391826377, 277G = 0. 9135791382, HK = 0. 9135791382B(r) 278 ELLIPTIC FUNCTIONK = 1. 9355810960, K' = 1. 7867691349, E = 1. 3055390943, E' = 1. 3931402485, r TABLE d = 50°9 = 0. 055019933698829, 6 0=0. 8899784604, HK = 0. 9715669451 279B(r) 28o ELLIPTIC FUNCTIONK = 2 . 0347153122, K' = 1 . 7312451757, E = 1 . 2586796248, E' = 1 . 4322909693, r TABLE e = 55°g = 0. 069042299609032, 6 0=0. 8619608462, HK = 1 . 0300875730B(r) 282 ELLIPTIC FUNCTIONK = 2. 1565156475, K' = 1. 6857503548, E = 1. 211056028, E' = 1.4674622093, r TABLE e = 60°q = 0. 085795733702195, 9 = 0. 8285168980, HK = 1. 0903895588 283B(r) 284 ELLIPTIC FUNCTIONK = 2. 3087867982, K' = 1. 6489952185, E = 1. 1638279645, E' = 1. 4981149284, r TABLE e = 65°q = 0. 106054020185994, 9 0=0. 7881449667, HK = 1. 1541701350 285B(r) 286 ELLIPTIC FUNCTIONK = 2. 5045500790, K' = 1. 6200258991, E = 1. 1183777380, E' = 1. 5237992053, ro TABLE d = 70°g = 0. 131061824499858, 6 = 0. 7384664407, HK = 1. 2240462555 287B(r) 288K = 2. 7680631454 = K'VS, K' = 1. 5981420021, E ELLIPTIC FUNCTION1 . 076405113, E' = 1 . 5441504969, r TABLE d = 75°q = 0. 163033534821580, 6 = 0. 6753457533, HK = 1. 3046678096B(r) 290 ELLIPTIC FUNCTIONK = 3. 1533852519, K' = 1. 5828428043, E = 1. 0401143957, E' = 1. 5588871966, r TABLE d = 80°g = 0. 206609755200965, G = 0. 590423578356, HK = 1. 406061468420 291B(r) 292 ELLIPTIC FUNCTIONK = 3. 2553029421, K' = 1. 5805409339, E = 1. 033789462, E' = 1. 5611417453, r TABLE 9 = 81°9 = 0. 217548949699726, G = 0. 5693797108, HK = 1. 4306906219 293B(r) 294 ELLIPTIC FUNCTIONK = 3. 3698680267, K' = 1. 5784865777, E = 1. 027843620, E' = 1. 5629622295, r TABLE e = 82°g =0. 229567159881194, 9 = 0. 5464169465, HK = 1. 4575481002 295B(r) 2q5 elliptic functionK = 3. 6004224992, K' = 1. 5766779816, E = 1.022312588, E' = 1. 5649475630, r TABLE = 83°q = 0. 242912974306665, 0=0. 5211317465, HK = 1. 4872214813 297B(r) 298 ELLIPTIC FUNCTIONK = 3.6518559695, K' = 1.5751136078, E = 1.017236918, E' = 1.5664967878, r T.ABLE e = 84°a = 0.257940195766337, 299e = . 4929628191, HK = 1 . 5205617314B(r) 300 ELLIPTIC FUNCTIONK = 3.8317419998, K' = 1. 5737921309, E = 1.0126635062, E' = 1.5678090740, r TABLE d = 85°9 = 0. 275179804873563, = 0. 4610905222, HK = 1. 5588714533 301B(r) 302 ELLIPTIC FUNCTIONK = 4. 0527581695, K' = 1. 5727124350, E = 1. 0086479569, E' = 1. 5688837196, r TABLE d = 86°q =0. 295488385558687, 0=0. 4242361430, HK = 1. 6043008048 303B(r) 304 ELLIPTIC FUNCTIONK = 4.3386539760, K' = 1.5718736105, E = 1.0052585872, E' = 1.5697201504, r TABLE, e = 87° .g = 0. 320400337134867, 9 0=0. 3802048484, HK = 1. 6608093153 305B(r) 3o6 ELLIPTIC FUNCTIONK = 4. 7427172653, K' = 1. 5712749524, E = 1. 0025840855, E' = 1. 5703179199, r TABLE d = 88°g = 0. 353165648296037, 9 0=0. 3246110213, HK = 1. 7370861537 307B(r) 3o8 ELLIPTIC FUNCTIONK = 5. 4349098296, K' = 1. 5709159581, E = 1. 0007515777, E' = 1. 5706767091, r TABLE = 89°q =0.403309306338378, 6 0=0.2457332317, HK = 1. 309 INDEXThe numbers refer to pages.PAGEAbsolute convergence 109Addition formulas, Elliptic Functions 250Algebraic equations 2Algebraic identities iAlternating series noArchimedes, spiral of 52Area of polygon 36Arithmetical progressions 26Asymptotes to plane curves 40Axial vector 95BBer and Bei functions 204BernouUian numbers 25polynomial 140Bessel functions 196addition formula 199multiplication formula 199references 213Bessel-Cliiford differential equation . . 205Beta functions 132Binomial coefficients 19Binormal 59Biquadratic equations 10Bromwich's expansion theorem 212 Cassinoid S3Catenary 52Cauchy's test 109Center of curvature, plane curves 39surfaces 56Change of variables in multiple inte-grals 17Characteristic of surface 56Chord of curvature, plane curves 39Circle of curvature 39Circular functions, see TrigonometryCissoid 5:^Clairaut's differential equation 166Coefficients, binomial 19Combinations 17Comparison test 109 Complementary function 167Concavity and convexity of planecurves 38, 42Conchoid 53Conditional convergence 109Confluent hypergeometric function. ... 185Conical coordinates 104Consistency of linear equations 15Convergence of binomial series 117tests for infinite series 109Covariant property 17Cubic equations 9Curl 93Curvature, plane curves 38space curves 58Curves, plane 36space 57Curvilinear coordinates 99Curvilinear coordinates, surfaces ofrevolution 106Cycloid 51Cyhndrical coordinates 32, 102Cylinder functions, see Bessel functions 197Dd'Alembert's Test 109Definite integrals, computation by dif-ference functions 225Simpson's method 221expressed as infinite series 134de Moivre's theorem 66Derivatives 155of definite integrals 156of implicit functions 161Descartes' rule of signs 5Determinants 11Difference functions 222Differential equations 162numerical solution 220Differentiation of determinants 13Discriminant of biquadratic equa-tion IIDivergence 93Double periodicity of elliptic functions 250311 312 INDEXE PAGEEllipse 46Ellipsoidal coordinates 102Elliptic cylinder coordinates 104Elliptic integrals, first kind 245second kind 248third kind 251Elliptic integral expansions 135, 195Envelope 40Envelope of surfaces 56Epicycloid 52Equations, algebraic 2transcendental, roots of 84Equiangular spiral 53Eta functions 251Euler's constant 27summation formula 25transformation formula 113theorem for homogeneous functions. 157Eulerian angles 32Evolute 39Exact differential equations 163, 177Expansion of determinants 13Expansion theorem, Bromwich's 212Heaviside's 212 Homogeneous differential equations162, 166, 177Homogeneous linear equations 15Horner's method 7I'Hospital's rule 145Hyperbola 48Hyperbolic functions 71spiral 52Hypergeometric differential equation 209series 209Hypergeometric function, confluent. . . 185Hypocycloid 52 Identities, algebraic iImplicit functions, derivatives of 161Indeterminate forms 145Indicial equation 174Infinite products 130series 109Integrating factors 163Interpolation formula, Newton's 22Intrinsic equation of plane curves. ... 44Involute of plane curves 39 Finite differences and sums 20Finite products of circular functions. . 84Finite series, special 26Fourier's series 136Fresnel's integrals 134Functional determinants 16 JJacobian 16 Ker and Kei functions 205Rummer's transformation 114 Gamma function 131Gauss's n function 133theorem 95Geometrical progressions 26Gradient of vector 93Graeffe's method 8Green's theorem 95Gregory's series 122Gudermannian 76HHarmonical progressions 26Harmonics, zonal 191Heaviside's operational methods 210expansion theorem 212Helical coordinates 106Hessian 16 Lagrange's theorem 112Laplace's integrals 193Latus rectum, ellipse 48hyperbola 49parabola 46Leclert's transformation 115Legendre's equation 191Leibnitz's theorem 157Lemniscate 53Limiting values of products 152sums 151Linear equations 15Linear vector function 96Lituus S3Logarithmic spiral S3MMaclaurin's theorem 112Markoff's transformation formula. ... 113 313PAGEMaxima and minima 152Mehler's integrals i93Minor of determinant 14Multinomial theorem 120Multiplication of determinants 12Multiple roots of algebraic equations . . 5NNeoid S3Neumann's expansion, zonal har-monics 194Newton's interpolation formula 22method for roots of equations 7theorem on roots of algebraic equa-tions 2Normal to plane curves 36Numbers, BernoulH's 140Euler's 141Numerical series 140Numerical solution of differential equa-tions 220 Oblate spheroidal coordinates 107Operational methods 210Orthogonal curvilinear coordinates 100 n function, Gauss's 133Parabola 45Parabolic coordinates 107Parabolic cylinder coordinates 105Parabolic spiral 53Parallelepipedon, volume of 92Partial fractions 20Particular integral 167Pedal curves 40Pendulum 247Permutations and combinations 17Plane 53Plane curves 36polar coordinates 41Plane geometry 34Points of inflexion 39. 42Polar coordinates 32, loiPlane curves 41Polar subtangent 37subnormal 2)7normal 37tangent 37Polar vector 95 Polynomial 2BernouUian 25series 119Principal normal to space curves 58Products, finite of circular functions. . 84limiting values of 152of two series noProgressions 26Prolate spheroidal coordinates 107 Quadratic equationsQuadriplanar coordinates . Raabe's test 109Radius of curvature, plane curves. . . 38, 42space curves 58surfaces 55Radius of torsion 59Reciprocal determinants 14Resolution into partial fractions 20Reversion of series 116Rodrigues' formula 193Roots of algebraic equations 2transcendental equations 84Rot 93Routh's rule 6 Scalar product 91Schlomilch's expansion, Bessel func-tions 201Series, finite, circular functions 81infinite 109special finite 26numerical 140of Bessel functions 201hypergeometric 209of zonal harmonics 194Simpson's method 221Singular points 41Skew determinants 14Skew-symmetrical determinants 15Solid geometry 53Space curves 57Spherical polar coordinates loiSpherical triangles 78Spheroidal coordinates 107Spiral of Archimedes 52Stirling's formula 28 314 INDEX Stokes's theorem 95Sturm's theorem 6Subnormal 3^Subtangent 36Sums, limiting values of 151Summation formula, Euler's 25Surfaces 55Symbolic form of infinite series 112Symbolic methods in differential equa-tions 173Symmetrical determinants 14Symmetric functions of roots ofalgebraic equations 2 Tables, binomial coefficients 20hyperbolic functions 72trigonometric functions 62Tangent to plane curves 36Taylor's theorem iiiTheta function 248, 251Toroidal coordinates 108Tractrix 53Transcendental equations, roots of 84Transformation of coordinates 29determinants 12 equations 4infinite series 113Triangles, solution of plane 77spherical 78Trigonometry 61Trilinear coordinates 33Trochoid 51UUniform convergence noUnit vector 92 Variation of parameters 180Vectors, axial 95polar 95functions, linear 96Vector product 91WWitch of Agnesi. 53 Zeta function 255Zonal harmonics 191