Exponential Integral Representations of Theta Functions

Let _3 (z):= ∑ _n∈ℤexp (iπ n^2 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, where ℝ_>0:= (0, +∞ ) . We define its logarithm log _3(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 log _3 (z) when z belongs to the hyperbolic quadrilateral ℱ^ ||_□:= {z∈ℂ | Im z> 0, -1≤Re z ≤ 1, |2 z - 1|> 1, | 2 z + 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 _4 and _2 may be conveniently expressed in terms of log _3 via functional equations, and hence get controlled as well. Our approach is based on a study of the logarithm of the Gauss hypergeometric function for a specific choice of the parameters. This has connections with the study of the universally starlike mappings introduced by Ruscheweyh, Salinas, and Sugawa.