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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)1-2e-scosh(τs)+e-2s,
f2(s,τ) =cos2(12s)1-2eiτscoss+e2iτs.

Then

23.11.2 (z)=1z2+80s(e-ssinh2(12zs)f1(s,τ)+eiτssin2(12zs)f2(s,τ))ds,

and

23.11.3 ζ(z)=1z+0(e-s(zs-sinh(zs))f1(s,τ)-eiτs(zs-sin(zs))f2(s,τ))ds,

provided that -1<(z+τ)<1 and |z|<τ.