Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ=x+iy, y > 0), such that f(τ+λ) = f(τ) where λ is real and f(-1/τ) = γ(-iτ)k f(τ), k being a complex number. The function (—iτ)k is defined as ek log(-iτ) where log(—iτ) has the real value when — iτ is positive and γ is a complex number with absolute value 1. Such functions have been studied by E. Hecke [4] who calls them functions with signature (λ, k, γ). We further assume that f(τ) = O(|y| -c) as y tends to zero uniformly for all x, c being a positive real number. It then follows that f(τ) has a Fourier expansion of the type f(τ) = a0 + Σ an exp(2πinτ/λ) (n = 1,2,…), the series being convergent absolutely in the upper half plane.