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Exponential Integral Representations of Theta Functions
Computational Methods and Function Theory ( IF 0.6 ) Pub Date : 2020-07-29 , DOI: 10.1007/s40315-020-00332-x
Andrew Bakan , Håkan Hedenmalm

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.



中文翻译:

Theta函数的指数积分表示

\(\ varTheta _ {3}(z):= \ sum \ nolimits _ {n \ in \ mathbb {Z}} \ exp(\ mathrm {i} \ pi n ^ 2 z)\)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z属于双曲四边形时\(\ log \ varTheta _ {3}(z)\)

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

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\(\ varTheta _ {4} \)\(\ varTheta _ {2} \)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。

更新日期:2020-07-29
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