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23 Weierstrass Elliptic and Modular FunctionsModular Functions

§23.17 Elementary Properties

Contents

§23.17(i) Special Values

23.17.1 λ(i) =12,
λ(eπi/3) =eπi/3,
23.17.2 J(i) =1,
J(eπi/3) =0,
23.17.3 η(i) =Γ(14)2π3/4,
η(eπi/3) =31/8(Γ(13))3/22πeπi/24.

For further results for J(τ) see Cohen (1993, p. 376).

§23.17(ii) Power and Laurent Series

When |q|<1

23.17.4 λ(τ)=16q(1-8q+44q2+),
23.17.5 1728J(τ)=q-2+744+1 96884q2+214 93760q4+,
23.17.6 η(τ)=n=-(-1)nq(6n+1)2/12.

In (23.17.5) for terms up to q48 see Zuckerman (1939), and for terms up to q100 see van Wijngaarden (1953). See also Apostol (1990, p. 22).

§23.17(iii) Infinite Products

23.17.7 λ(τ)=16qn=1(1+q2n1+q2n-1)8,
23.17.8 η(τ)=q1/12n=1(1-q2n),

with q1/12=eiπτ/12.