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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.8 Trigonometric Series and Products

Contents

§23.8(i) Fourier Series

If q=πω3/ω1, (z/ω1)<2(ω3/ω1), and z𝕃, then

23.8.1 (z)+η1ω1-π24ω12csc2(πz2ω1) =-2π2ω12n=1nq2n1-q2ncos(nπzω1),
23.8.2 ζ(z)-η1zω1-π2ω1cot(πz2ω1) =2πω1n=1q2n1-q2nsin(nπzω1).

§23.8(ii) Series of Cosecants and Cotangents

When z𝕃,

23.8.3 (z)=-η1ω1+π24ω12n=-csc2(π(z+2nω3)2ω1),
23.8.4 ζ(z)=η1zω1+π2ω1n=-cot(π(z+2nω3)2ω1),

where in (23.8.4) the terms in n and -n are to be bracketed together (the Eisenstein convention or principal value: see Weil (1999, p. 6) or Walker (1996, p. 3)).

23.8.5 η1=π22ω1(16+n=1csc2(nπω3ω1)),

with similar results for η2 and η3 obtainable by use of (23.2.14).

§23.8(iii) Infinite Products

23.8.6 σ(z)=2ω1πexp(η1z22ω1)sin(πz2ω1)n=11-2q2ncos(πz/ω1)+q4n(1-q2n)2,
23.8.7 σ(z)=2ω1πexp(η1z22ω1)sin(πz2ω1)×n=1sin(π(2nω3+z)/(2ω1))sin(π(2nω3-z)/(2ω1))sin2(πnω3/ω1).