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23 Weierstrass Elliptic and Modular FunctionsWeierstrass Elliptic Functions

§23.12 Asymptotic Approximations

If q(=eπiω3/ω1)0 with ω1 and z fixed, then

23.12.1 (z)=π24ω12(-13+csc2(πz2ω1)+8(1-cos(πzω1))q2+O(q4)),
23.12.2 ζ(z)=π24ω12(z3+2ω1πcot(πzω1)-8(z-ω1πsin(πzω1))q2+O(q4)),
23.12.3 σ(z)=2ω1πexp(π2z224ω12)sin(πz2ω1)(1-(π2z2ω12-4sin2(πz2ω1))q2+O(q4)),

provided that z𝕃 in the case of (23.12.1) and (23.12.2). Also,

23.12.4 η1=π24ω1(13-8q2+O(q4)),

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