Exponential Integral Representations of Theta Functions

Abstract

Let $\varTheta _{3} (z):= \sum \nolimits _{n\in \mathbb {Z}} \exp (\mathrm {i} \pi n^2 z)$ be the standard Jacobi theta function, which is holomorphic and zero-free in the upper half-plane $\mathbb {H}:=\{z\in \mathbb {C}\,|\,\,\mathrm{{Im}}\, z>0\}$, and takes positive values along $\mathrm {i} \mathbb {R}_{>0}$, the positive imaginary axis, where $\mathbb {R}_{>0}:= (0, +\infty )$. We define its logarithm $\log \varTheta _3(z)$ which is uniquely determined by the requirements that it should be holomorphic in $\mathbb {H}$ and real-valued on $\mathrm {i} \mathbb {R}_{>0}$. We derive an integral representation of $\log \varTheta _{3} (z)$ when z belongs to the hyperbolic quadrilateral

$$\begin{aligned} \mathcal {F}^{\,{{{||}}}}_{{{{\square }}}}:= \big \{z\in \mathbb {C}\ \big | \ \,{\mathrm{{Im}}}\, z> 0, \,\,-1\le {\mathrm{{Re}}}\, z \le 1, \,\,|2 z\! -\! 1|> 1,\,\, | 2 z \!+\! 1|\! > 1\big \}. \end{aligned}$$

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 $\varTheta _{4}$ and $\varTheta _{2}$ may be conveniently expressed in terms of $\log \varTheta _{3}$ 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.