# §20.10 Integrals

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

 20.10.1 $\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $\Re$: real part Keywords: Mellin transform Referenced by: §20.10(i), Erratum (V1.1.5) for §20.10(i) Permalink: http://dlmf.nist.gov/20.10.E1 Encodings: TeX, pMML, png Modification (effective with 1.1.5): Originally the constraint was $\Re s>2$. This can be relaxed to $\Re s>1$ because $\theta_{2}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{4}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5). See also: Annotations for §20.10(i), §20.10 and Ch.20
 20.10.2 $\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>1$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $\Re$: real part Keywords: Mellin transform Referenced by: §20.10(i), Erratum (V1.1.5) for §20.10(i) Permalink: http://dlmf.nist.gov/20.10.E2 Encodings: TeX, pMML, png Modification (effective with 1.1.5): Originally the constraint was $\Re s>2$. This can be relaxed to $\Re s>1$ because $\theta_{3}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{3}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4). See also: Annotations for §20.10(i), §20.10 and Ch.20
 20.10.3 $\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),$ $\Re s>0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$: theta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential, $\mathrm{i}$: imaginary unit, $\int$: integral and $\Re$: real part Keywords: Mellin transform Referenced by: §20.10(i), Erratum (V1.1.5) for §20.10(i) Permalink: http://dlmf.nist.gov/20.10.E3 Encodings: TeX, pMML, png Modification (effective with 1.1.5): Originally the constraint was $\Re s>2$. This can be relaxed to $\Re s>0$ because $1-\theta_{4}\left(0\middle|ix^{2}\right)=1-x^{-1}\theta_{2}\left(0\middle|ix^{% -2}\right)\sim 1$ as $x\to 0+$, which follows from (20.7.33) and (20.4.3). See also: Annotations for §20.10(i), §20.10 and Ch.20

Here $\zeta\left(s\right)$ 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 $\left|\Re\beta\right|+\left|\Im\beta\right|\leq\ell$. Then

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

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).