Exponential integral representations of theta functions

TL;DR

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Contribution

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Abstract

Let $\Theta_{3} (z):= \sum_{n\in\mathbb{Z}} \exp (i \pi n^2 z)$ be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane $\mathbb{H}$, and takes positive values along the positive imaginary axis. We define its logarithm $\log\Theta_3(z)$ which is uniquely determined by the requirements that it should be holomorphic in $\mathbb{H}$ and real-valued on the positive imaginary axis. We derive an integral representation of $\log\Theta_{3} (z)$ when $z$ belongs to the hyperbolic quadrilateral $\mathcal{F}^{||}_{\square}$, consisted of all those $z \in \mathbb{H}$ which satisfy $-1 \leq Re\, z \leq 1$, $|2 z - 1| > 1$ and $ |2 z + 1| > 1$. Since every point of $\mathbb{H}$ is equivalent to at least one point in $\mathcal{F}^{||}_{\square}$ under the theta subgroup of the modular group on the upper half-plane, this representation carries over in modified form to all of $\mathbb{H}$ via the identity recorded by Berndt. The logarithms of the related Jacobi theta functions $\Theta_{4}$ and $\Theta_{2}$ may be conveniently expressed in terms of $\log\Theta_{3}$ 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.