704]
29. We have
A MEMOIR ON THE SINGLE AND DOUBLE THETA-FUNCTIONS.
The constants of the theory.
AO = 1 + 2q + 2q i + 2g 9 +...,
477
B0= 2q? + 2q î + 2q™ +
CO = 1 — 2q + 2q i — 2q 9 + ...,
DO = 0,
D' 0= — Tr{(f — + 5q ? ^ — ...}.
If, as definitions of k, k', K, we assume
J № C 2 0 _J4) m
A 2 0’ ^ ~A 2 0’ £0 ' CO ’
then we have
k =4 Vq
- f 1 + q 2 + q 6 + ...
[1 + 2q + 2 ç 4 + ...J
, =4 Vg(l — 4>q+ 14g 2 + ...),
k' =
ii+tw+fiÎ . = 1 - 8 Ï + - 96ÿ + ....
jsf _,r(l + 2 g + 2g« + ...)(l-3ÿ + 5g'-...) , „ + 4 „. ,<>„» + ,
2(1-2j + 25*-...)(1 + 9 s + ?‘+...) ’ 1^0 + »? + «« +°2 + ••••>’
where I have added the first few terms of the expansions of these quantities. We
have identically
k 2 + k' 2 = 1.
It will be convenient to write also, as the definition of E,
C"0
K (K — E) =
CO *
we have then
E = K-l C "°
moreover,
giving
and thence
30. Other formulæ are
k = 4
K CO ’ AO. BO .CO. D'O ‘ A2 ° + 520 • 00 • ’
E _ 1 C"0 _ 2it 2 q — 4ç 4 + 9ç 9 — ...
~K~ K 2 ‘ <70 ’ ’ = if 7 ' T - 2q + 2q*+ ... 5
~ = 1 -8q + 4<8q 2 - 224 i q 3 + ...,
E = \ r n {1 — 4g + 20g 2 — 64ÿ 3 +...}.
- (1 + q 2 .1 +g ,4 ...| 4
k' =
1+ q. 1 + q 3 ..
11 — q . 1 — q 3 .,
(1 + q . 1 + q 3 ...j ’
[1 + q . 1 + q 3 ... 1 — q 2 . 1 — q* ...} 2
[1 — q . 1 — q 3 ... 1 4- q 2 . 1 + q 4 ...}