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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.8 Trigonometric Series and Products

  1. §23.8(i) Fourier Series
  2. §23.8(ii) Series of Cosecants and Cotangents
  3. §23.8(iii) Infinite Products

§23.8(i) Fourier Series

If q=eiπω3/ω1, (z/ω1)<2(ω3/ω1), and z𝕃, then

23.8.1 (z)+η1ω1π24ω12csc2(πz2ω1) =2π2ω12n=1nq2n1q2ncos(nπzω1),
23.8.2 ζ(z)η1zω1π2ω1cot(πz2ω1) =2πω1n=1q2n1q2nsin(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=112q2ncos(πz/ω1)+q4n(1q2n)2,
23.8.7 σ(z)=2ω1πexp(η1z22ω1)sin(πz2ω1)×n=1sin(π(2nω3+z)/(2ω1))sin(π(2nω3z)/(2ω1))sin2(πnω3/ω1).