Theta Functions

Abstract

Theta functions provide another source of elliptic functions as quotients of theta functions. They are defined for a lattice L of the form L _τ = Zτ + Z with lm(τ) > 0. This is no restriction, because every lattice L is equivalent to some L _τ. Since these functions f(z) are always periodic f(z) = f(z + 1), we will consider their expansions in terms of q _z = e ^2piz where f(z) = f*(q _z) and f* is defined on C* = C — {0}. In §1 we consider various expansions in the variable q = q _z of functions introduced in the previous chapter.