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

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.

令\（\ varTheta _ {3}（z）：= \ sum \ nolimits _ {n \ in \ mathbb {Z}} \ exp（\ mathrm {i} \ pi n ^ 2 z）\）为标准Jacobi theta函数，在上半平面是全纯的且无零\（\ mathbb {H}：= \ {z \ in \ mathbb {C} \，| \，\，\ mathrm {{Im}} \ ，z> 0 \} \），并沿正虚轴\（\ mathrm {i} \ mathbb {R} _ {> 0} \）取正值，其中\（\ mathbb {R} _ {> 0}：=（0，+ \ infty）\）。我们定义其对数\（\ log \ varTheta _3（z）\），该对数由以下要求唯一确定：在\（\ mathbb {H} \）中应为全纯，而在\（\ mathrm {i}中应为实值\ mathbb {R} _ {> 0} \）。我们推导了当z属于双曲四边形时\（\ log \ varTheta _ {3}（z）\）

由于\（\ mathbb {H} \）的每个点至少等于\（\ mathcal {F} ^ {\，{{{||}}}} __ {{{{\ square}} }} \）在上半平面的模块化组的theta子组下，该表示形式通过Berndt记录的身份以修改后的形式延续到\（\ mathbb {H} \）的所有形式。相关的Jacobi theta函数\（\ varTheta _ {4} \）和\（\ varTheta _ {2} \）的对数可以方便地用\（\ log \ varTheta _ {3} \）表示。通过功能方程，因此也可以得到控制。我们的方法基于对高斯超几何函数的对数的研究，用于特定的参数选择。这与Ruscheweyh，Salinas和Sugawa引入的普遍星状图的研究有关。