Check formula: 



KE' + K'E-KK' = 



For reader's convenience 



TT =3.14159 26535 89793 23846 

 TT^ =9.86960 44010 89358 62 



^ = 1.57079 63267 94896 61923 



(A) 

 sn(«, k) = 



cn(M, k) = 



Os(q) Oi(v,q) 

 e,(q) eo(v,q) 



l+2q + 2q' + 2q'' + . 

 l+q^ + q'^ + q^^ + . Ll — 2g cos 27rz/ + 2g* cos 47rz/ — 2g'' cos 677^ . 



'sin irv — q^ sin Sirv+q''' sin Sirv—q'^- sin 77rv 



6o_iq) 6o{v,q) 

 _l-2q + 2q^-2q^ 



"cos -rrV+q^ COS SirV+q^ COS SttV +q'^^ COS 7ttV + .' 



\+q' + q^ + q'^~ . L 1 — 2g cos 27rt/ + 2g* cos 47rz/— 2g^ cos 67rt; + . _ 



j^/„ N_ ^q(^) Oz(v< q) 

 an ( u, K ) = -—- — ^ • — — — 



_ l—2q + 2q^ — 2q^ . T 1 +2(7 cos 27rt;+2^* cos 47rz/ + 2(7'' cos 67rz/+. 



l+2q + 2q* + 2q' 



:[\^- 



2q cos 27rz/ + 2(7* cos 477^ — 2^^ cos 67rZ'+. _ 



where v= -^-pr . There are similar expansions for sc(m k), sd(«, k), cd(« k). 

 These expansions imply similar ones for sn (u, k), en (w k), dn (u «') etc., 



where q' and v' are substituted for q and z' and ■y' = 



2K' 



Also these expansions are true for the imaginary argument in. 

 Thus : 



(B) 



— i sn(iu, k) = snh (n, k) 

 Os(q) 



sinh TTV — q- sinh 2)TTV+q^ sinh Sirv — q^- sinh 77ri^ + . . 1 

 ^2(^7) L 1 — 2g cosh 27rz/ + 2g* cosh 47rt/ — 2g^ cosh 6ttv^- . . . J 



cn(w, k) = cnh (h, k) 



_ ^o(<7) f cosh ?;-?/ + g^ cosh 3Tr^'4-<7" cosh Sirv+q'^^ cosh 77rz^+. . 

 ~ ^2(9) L 1— 2g cosh 277-1/ + 2g* cosh 4x^^ — 2(7'' cosh 677^ + . . . 



dn(ni, k)= dnh(z(, k) 



e-Aq) 



1+2(7 cosh 27r7; + 2(7^ cosh Airv + 2q^ cosh 67rz;+. 

 1— 2g cosh 27rz/+2g* cosh 47r^' — 2(7*' cosh 671-2' + . 



where v = 



2K 



dn 



These, of course, imply similar expansions for —is>n{iii, k), cn(hi, k') and 



w 

 (lit, k') where q' and v' are sul)stituted for v and 5 and v'= "o^' 



360 



