C A ] 2 3 D ec 2 01 9 Exponential integral representations of theta functions

Let Θ3(z) := ∑ n∈Z exp(iπn z) be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane H := {z ∈ C | Im z > 0}, and takes positive values along iR>0, the positive imaginary axis. We define its logarithm logΘ3(z) which is uniquely determined by the requirements that it should be holomorphic in H and real-valued on iR>0. We derive an integral representation of logΘ3(z) when z belongs to the hyperbolic quadrilateral F || := { z ∈ C : Im z > 0, −1 ≤ Re z ≤ 1, |2z−1| > 1, |2z+1|> 1 } . Since every point of H is equivalent to at least one point in F || under the theta subgroup of the modular group on the upper half-plane, this representation carries over in modified form to all of H via the identity recorded by Berndt. The logarithms of the related Jacobi theta functions Θ4 and Θ2 may be conveniently expressed in terms of logΘ3 via functional equations, and hence get controlled as well. Our approach is based on a study the logarithm of the Gauss hypergeometric function for a specific choice of the parameters. This connects with the study of the universally starlike mappings introduced by Ruscheweyh, Salinas, and Sugawa.