-35-
Expansions of the .9 and elliptic functions,

In practice, it is often easier in expanding a product of
the type of M on page 26, to expand each infinite product into a
series and then multiply the series as descrited above. For example,
expand 9o(s) = II(1 - ¢*") II(1 ~ 2¢°™ o822 + ¢*™%). Letting ¢® = M
this can be written .9,(2) =II(1 - ¢™") II(1 ~ t%q%") II(1 - t=%%Y) ,
From the formulas on page 33
II(1 =~ g™) = 1= g% = ¢* 4 g% 4 soveee,
TI(1 - t%q®™2) = 1 = £% = £%¢® 4 2%¢% = 1255 4 $49° = t297 + 24" 4 *+o0e,
II(1 = £°%%3) = 1 = £729 — 7248 4 1™4g% ~ £73¢f 4 t74¢° - %7 4 A4 een
Multiplying these series, using the results of page 34,
So(2) =1~ g(t2+ %) + *(t* % =1) + P2+t 2- 2=tV

+ g (= BB 2R o] 4 g4 g 7YY e
=1-2qcos 22 + 2¢*cos 4z = *reee

In the same manner we obtain
Oy (2) = T1(X ~ ¢*™)II(1 + 2¢*™ *cos 22 + ¢*™?) = 1 + 29 cos 22 + 2¢° cos 4z + **°,
By (2) = 21* sinz I1(1 = ¢*") 1I(1 -2¢°" cos 22 + ¢*")

- q*(t =Y 111 - ¢®) T1(1 - £%%) 111 ~ £~%*")

1
=2q [8in - ¢®sin 3z + q° sin 5z ceesess 155

Pa(e) = 20* cos e (1~ q®) II(1 +2¢* cos 2z + ¢*M)
= q'}(t + £ IQ - o) 111 + t2%°") QA + £7%*™)

=2q* [cosl % q° cos 32 + q‘ cos S5z + o-.,..] 5

The product II(1 1 2¢*" cos @z + ¢*") can be expanded without
factoring by using the results of page 34. Llet x= g%
op=32cos 2, and by=1. Then
I1(1 4 2¢°%0s 22 4 ¢*") = 1 4 29%c0s 22+ (1 4 2cos 22)¢* + (4cos™2e $2co8 28)q% e,
Using this form of the series, the series are obtained in powers
of cosines of 22, instead of cosines of multiples of 2a. Forexample,
A,(2) = Zqisin 21~ ¢*(142 cos 2z) = ¢%1 +2 cos 2z - 4 cos®2s) sevee },
O;(2) = 2qicos 21~ ¢*(1-2cos 22) -~ ¢%(1 + 2 cos 22 ~ 4 cos®2a) *+ver },