SMITHSONIAN MISCELLANEOUS COLLECTIONS V26

A" SMITHSONIAN MISCELLANEOUS COLLECTIONS VOLUME 109 (WHOLE VOLUME) Eesearcf) Corporation Jfunb SMITHSONIAN ELLIPTIC FUNCTIONS TABLES Prepared by G W AND R M SPENCELEY Miami University, Oxford, Ohio (Publication 3863) CITY OF WASHINGTON PUBLISHED BY THE SMITHSONIAN INSTITUTION NOVEMBER L 1947 PREFACE These tables were inspired by the similar work of Sir George Greenhill and Col R L Hippisley as published in Smithsonian Mathematical Formulae and Tables of Elliptic Functions The pattern they designed has been followed closely by us The chief difference is our inclusion of the three elliptic functions, sn(u,k), cn(u,k), dn(u,k) Columns A and D were computed first From them were computed columns sn(u) and cn(u), checked by means of Check I: sn^(u) +cn-(u) =1, with a maximum error of ±2 in the 15th decimal Practically all errors in these first four columns were eliminated at this time, but some few managed to elude us Column dn(u) was then computed, and checked by means of Check II: sn(K— u) dn(u) =:cn(u), with a maximum error of ±2 in the 15th decimal Those few errors that had eluded us in Check I were discovered at this time Column E(u,k) followed, computed by the formula E(u + v)=E(u)+E(v)— k2sn(u+v)sn(u)sn(v), where u was taken as r/90 i.e 1° K and tabulated r°, while v was taken as 1/90 K, Independent computations for r° = 15°, 30°, 45°, 60°, 75°, 90° were maximum used as Check III, with a divergence of ±3 in the 15th decimal In this method of computing the E(u,k) column the errors are additive, and is more sensitive than Checks I and II The ^ column was computed last by means of sin ^ = sn(u), using Andoyer's hence Check III and Cosines This was checked by computing 15-place Table of Natural Sines cos 94 39 10 00 11 41 11 01 06 09 18 6Z 62 13 78 80 : APPENDIX In the Legendre-Jacobi-Abel system for computing compiled from the following list functions, the elliptic following formulas, and tables, will be found useful The formulas were of reference texts Smithsonian Mathematical Formulae, Edwin P Adams, Ph.D of Elliptic Functions, Harris Hancock, Ph.D Elliptic Functions, Alfred Cordew Dixon, M.A Functions of Complex Variable, James Pierpont, L.L.D Advanced Calculus, Edwin Bidwell Wilson, Ph.D Theory Our use of the notations snh (u, k), en h (m, k), dnh (u, k) in the formulas may below be strange to the reader, but they seem to be a natural extension of the hyperbolic notation to the sn (u, k), en (u, k), dn (u, k) We tions have used them for years we hope ; func- elliptic they will be acceptable to other computers The angle number q may be computed number / is computed from as follows: basic 6, the given the modular - V cos whence + Vcos^ ^=/+2/5 + i5/9 + l50P + 1707F + 20910P+268616P + , / For 16-place accuracy for these we may = 4^ , 61 , , becomes this difficult, if functions Jacobi ej^v, = l+2q q)=l-2q not impossible, for = 6>70° 7r^ : ) sin 37rt/+g« sin STrt/-^'^ sin 7-nv+ .) cos 37rv + q^ cos SiTV+q^^ cos 7ttv + cos 27rt/+25* cos 47rt/+2g« cos 67rt;+2g" cos 87rt;+ cos 27n; + 2q* cos 4^-2q'> cos 67rt/+2(7" cos 27rt/- e^(v,q)=2qi(sm ifv-q^ e^lv, q) =2qilcos -rv+q"^ Oz{v q) use the relation In q-ln g' The , • • • whence do(q)=2qHl+q' + q' + q'' + q'' + q'° + q*' + To compute K, K' we •) • - = l-{-2q + 2q' + 2q' + 2q'' + 2q" + 2q'' + 2q''- eo(q) = l-2q + 2q*-2q' + 2q''-2q'' + e,{q) • • have K' -^ ^ =dsiq), and, since q = e To compute ^ E, E' £'_ , it K'= follows that In q we have 2,r= r q-Aq* + 9q'-\6q^' + 2Sq''- ^~~K-~K^\l-2q + 2q*-2q' + 2q''-2q'' + 359 "1 J ; — : Check formula: KE' + K'E-KK' = For reader's convenience =3.14159 26535 89793 23846 =9.86960 44010 89358 62 TT TT^ ^ = 1.57079 63267 94896 61923 (A) sn(«, k) = Os(q) Oi(v,q) e,(q) eo(v,q) l+2q + 2q' + 2q'' + l+q^ + cn(M, k) q'^ + q^^ sin Sirv+q''' sin Sirv—q'^- sin 77rv 6o{v,q) \+q' + q^ + q'^~ K — q^ + Ll — 2g cos 27rz/ + 2g* cos 47rz/ — 2g'' cos 677^ _l-2q + 2q^-2q^ ( 'sin irv = 6o_iq) j^/„ an u, N_ = ) ^q(^) -—^ — • —Oz(v< "cos -rrV+q^ COS L SirV+q^ COS ' + SttV +q'^^ COS 7ttV — 2g cos 27rt/ + 2g* cos 47rz/— 2g^ cos 67rt; + _ q)— cos 67rz/+ _ l—2q + 2q^ — 2q^ T +2(7 cos 27rt;+2^* cos 47rz/ + l+2q + 2q* + 2q' :[\^- 2q cos 27rz/ + 2(7* cos 477^ — 2^^ cos 67rZ'+ 2(7'' where v= -^-pr There are similar expansions for sc(m These expansions imply similar ones for sn where q' and are substituted for q and v' z' (u, and y' k), sd(«, k), cd(« k), en (w k), dn (u _ k) «') etc., = 2K' Also these expansions are true for the imaginary argument in Thus (B) — i sn(iu, k) = snh (n, k) ^2(^7) cn(w, k) dn(ni, L cnh _ ~ ^o(n{iii, k), cn(hi, dn (lit, (h, k) e-Aq) where v = — 2g = k)= — q^- sinh 77ri^ + — 2g^ cosh 6ttv^47rt/ sinh TTV — q- sinh 2)TTV+q^ sinh Sirv Os(q) where w q' and v' are sul)stituted for 360 v and and v'= "o^' k') and — — — : ^ For values of sn(w, k), cn(u,K), dn{u, k) where the modular angle lies between tabulated values, and where interpolation fails, direct computation may be employed up to, and including, ^ = 89°, using equations (A) above Values, 15-places correct, can be thus computed without prohibitive labor That, in fact, was the way our tables were computed For values of between 89° and 90° we may resort to Jacobi's imaginary transformation equations (C) , sn(M, K)= — —isn(in,K') -~ -—^ = cn(M^ k) —— — ^~ cnh(M, dn(ni,fc') -^ Qn{l{, k)=: y- cnh (u,k') = cn{iu,K) J — snh(z/,K') = en {iu,k') k') —— dnh(jf,/c') = - ^^ cnh(M, cn(iM, /c) k') together with (D) sn(K — n, cn(u, k) k) dn(?/, k) cn(K — u, k', sn(!