Is there a possibility to transform the Jacobi theta function $$\theta_ 3 (e^{-\pi z (1 - x)})$$ in something like:$$ f[z] \theta_ 3 (e^{-\pi z x})$$ where f[z] is a function e.g. theta function only depending on z. Is the Dedekind eta function a possible ansatz?
Transformation of Jakobi theta function $\theta_ 3 (e^{-\pi z (1 - x)}) to f[z] \theta_ 3 (e^{-\pi z x})$
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functional-analysis
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0$\theta_3$ is a function of two variables $z\in \mathbb{C}$ and $\tau$ in the upper half plane. So what does your function express exactly? – 2017-01-05
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0You are right, the notation isn't good. Yesterday I posted an identity:$$\int_ 0^1\t heta_ 3 (0, e^{-\pi^2 t (1 - \tau)})\, \t heta_ 2 (0, e^{-\pi^2\tau})\ d\tau = 1 $$. I have a prove via composition theorem and I try to get more understanding of that integral, so I asked about this transform. – 2017-01-05
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0Sorry, I missed a "t" in the $\theta_2$ function and before the integral.$$t\, \int_ 0^1\t heta_ 3 (0, e^{-\pi^2 t (1 - \tau)})\, \t heta_ 2 (0, e^{-\pi^2 t\tau})\ d\tau = 1 $$ – 2017-01-05