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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.11 Integral Representations

Let τ=ω3/ω1 and

23.11.1 f1(s,τ) =cosh2(12τs)12escosh(τs)+e2s,
f2(s,τ) =cos2(12s)12eiτscoss+e2iτs.

Then

23.11.2 (z)=1z2+80s(essinh2(12zs)f1(s,τ)+eiτssin2(12zs)f2(s,τ))ds,

and

23.11.3 ζ(z)=1z+0(es(zssinh(zs))f1(s,τ)eiτs(zssin(zs))f2(s,τ))ds,

provided that 1<(z+τ)<1 and |z|<τ.