# §20.10 Integrals

## §20.10(i) Mellin Transforms with respect to the Lattice Parameter

Let $s$ be a constant such that $\Re{s}>2$. Then

 20.10.1 $\int_{0}^{\infty}x^{s-1}\mathop{\theta_{2}\/}\nolimits\!\left(0\middle|ix^{2}% \right)\mathrm{d}x=2^{s}(1-2^{-s})\pi^{-s/2}\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}s\right)\mathop{\zeta\/}\nolimits\!\left(s\right),$
 20.10.2 $\int_{0}^{\infty}x^{s-1}(\mathop{\theta_{3}\/}\nolimits\!\left(0\middle|ix^{2}% \right)-1)\mathrm{d}x=\pi^{-s/2}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}% s\right)\mathop{\zeta\/}\nolimits\!\left(s\right),$
 20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\mathop{\theta_{4}\/}\nolimits\!\left(0\middle|ix^{% 2}\right))\mathrm{d}x=(1-2^{1-s})\pi^{-s/2}\mathop{\Gamma\/}\nolimits\!\left(% \tfrac{1}{2}s\right)\mathop{\zeta\/}\nolimits\!\left(s\right).$

Here $\mathop{\zeta\/}\nolimits\!\left(s\right)$ again denotes the Riemann zeta function (§25.2).

For further results see Oberhettinger (1974, pp. 157–159).

## §20.10(ii) Laplace Transforms with respect to the Lattice Parameter

Let $s$, $\ell$, and $\beta$ be constants such that $\Re{s}>0$, $\ell>0$, and $\mathop{\sinh\/}\nolimits\left|\beta\right|\leq\ell$. Then

 20.10.4 $\int_{0}^{\infty}e^{-st}\mathop{\theta_{1}\/}\nolimits\!\left(\frac{\beta\pi}{% 2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{\infty}e^{-% st}\mathop{\theta_{2}\/}\nolimits\!\left(\frac{(1+\beta)\pi}{2\ell}\middle|% \frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=-\frac{\ell}{\sqrt{s}}\mathop{\sinh% \/}\nolimits\!\left(\beta\sqrt{s}\right)\mathop{\mathrm{sech}\/}\nolimits\!% \left(\ell\sqrt{s}\right),$
 20.10.5 $\int_{0}^{\infty}e^{-st}\mathop{\theta_{3}\/}\nolimits\!\left(\frac{(1+\beta)% \pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\int_{0}^{\infty}% e^{-st}\mathop{\theta_{4}\/}\nolimits\!\left(\frac{\beta\pi}{2\ell}\middle|% \frac{i\pi t}{\ell^{2}}\right)\mathrm{d}t=\frac{\ell}{\sqrt{s}}\mathop{\cosh\/% }\nolimits\!\left(\beta\sqrt{s}\right)\mathop{\mathrm{csch}\/}\nolimits\!\left% (\ell\sqrt{s}\right).$

For corresponding results for argument derivatives of the theta functions see Erdélyi et al. (1954a, pp. 224–225) or Oberhettinger and Badii (1973, p. 193).

## §20.10(iii) Compendia

For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).