Is there any way to simplify an expression like this $\sum_{ m = - \infty}^{\infty} e^{-am^2/2 + bm}$? I know there exist an identity for a similar expression, just integrating, does the same identity still hold for the summation case? If so, how can I argue that the identity still holds. Thanks!
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The fact that such sum has no known closed form leads to defining the Jacobi theta function. – Sangchul Lee Feb 08 '18 at 17:21
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The fact that an integral has a closed form, doesn't mean the related sum does. – Yuriy S Feb 08 '18 at 20:11
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Depends on exactly what you mean by "simplify". The summation is equal to $f(e^{b-a/2},e^{-b-a/2})$ where $f(,)$ is Ramanujan's general theta function. I don't know what identity you are referring to.
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