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I need to evaluate an integral that involves the Jacobi Theta Function $\vartheta_3(z,q)$ defined by MathWorld as $$\vartheta_3(z,q)=\sum_{n\in\mathbb{Z}}q^{n^2}e^{2inz}.$$ Specifically, I wish to evaluate $$\int_0^1 q^{s-1} \vartheta^2_3(0,q) dq.$$

If that is possible, I would also like to do the same for other powers of $\vartheta_3(q)$ $$\int_0^1 q^{s-1} \vartheta^n_3(0,q) dq.$$

In this answer, I see that it is possible to evaluate this integral for $n=1$ to be $$\int_0^1 q^{s-1} \vartheta_3(0,q) dq=\frac{\pi}{\sqrt{s}}\coth(\pi x)=\sum_{n\in\mathbb{Z}} \frac{(-1)^n \cos(\pi n)}{n^2+s}.$$ This was done by recognizing the Fourier Series $$\frac{\pi}{x}\coth(\pi x)=\sum_{n\in\mathbb{Z}} \frac{(-1)^n \cos(\pi n)}{n^2+x^2}.$$

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goodwitm
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  • have you tried to employ Ramanujan's Master theorem? – tired Apr 05 '17 at 07:10
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    this here might also be of some interest: http://math.stackexchange.com/questions/1811490/family-of-definite-integrals-involving-dedekind-eta-function-of-a-complex-argume – tired Apr 05 '17 at 07:12
  • I have not tried Ramanujan's Master theorem, but it seems to be used for integrals to $\infty$, and $\vartheta_3$ blows up at $q=1$. As for the other comment, I will try writing it with a complete elliptical integral of the first kind, though I see no easy way to write the $q^{s-1}$ in terms of the elliptic modulus. – goodwitm Apr 05 '17 at 20:00

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