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In my recent explorations I stumbled upon the following series

$$ \vartheta_{4}(0,e^{\alpha \cdot z})=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot z\cdot k^{2}} ; \alpha \in \mathbb{R}, z \in \mathbb{C} $$

This is one of the well known Jacobi theta functions/series with the peculiarity of having the variable $z \in \mathbb{C}$ in a different place, i.e. $e^{\alpha \cdot \mathbf{z} \cdot k^{2}}$!!

The usual form of the theta function is

\begin{align*} \vartheta_{4}(z,e^{\alpha })=1+2\sum_{k=1}^{\infty} (-1)^{k}\cdot e^{\alpha \cdot k^{2}}\cos(2kz) ; \end{align*}

but not in the case I have in hands. Does the former formula make any sense? Where are this kind of series used or analysed? (Apart from the well known case of

$$\psi(x)=\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2} \left[ \vartheta_{3}(0,e^{-\pi x})-1 \right]$$ used in the context of the Riemann zeta-function.)

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    For convergence we need $\mathrm{Im} (\alpha\cdot z) < 0$, right?2011-08-06
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    @GEdgar, thats right.2011-08-06
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    [I corrected your theta function formula.](http://dlmf.nist.gov/20.2.E4)2011-08-07

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