  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.

Classical Analysis and ODEs