20 Theta Functions Properties 20.9 Relations to Other Functions 20.11 Generalizations and Analogs

§20.10 Integrals

Keywords:: integrals, theta functions
Permalink:: http://dlmf.nist.gov/20.10

§20.10(i) Mellin Transforms with respect to the Lattice Parameter
§20.10(ii) Laplace Transforms with respect to the Lattice Parameter
§20.10(iii) Compendia

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

Notes:: For (20.10.1) and (20.10.3) use §20.7(viii) with appropriate changes of integration variable. For (20.10.2) use (20.2.3) with , $\tau=it$ , Bellman (1961, pp. 28–32), Koblitz (1993, pp. 70–75), and/or Titchmarsh (1986b, §2.6).
Keywords:: theta functions
Referenced by:: §20.9(iii)
Permalink:: http://dlmf.nist.gov/20.10.i

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

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, : differential of and $\int$ : integral
Referenced by:: §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E1
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, : differential of and $\int$ : integral
Referenced by:: §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E2
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, $\mathop{\zeta\/}\nolimits\!\left(s\right)$ : Riemann zeta function, : differential of and $\int$ : integral
Referenced by:: §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E3
Encodings:: TeX, pMML, png

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

Notes:: See Bellman (1961, pp. 20–24).
Keywords:: theta functions
Permalink:: http://dlmf.nist.gov/20.10.ii

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

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

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : differential of , : base of exponential function, $\mathop{\mathrm{sech}\/}\nolimits z$ : hyperbolic secant function, $\mathop{\sinh\/}\nolimits z$ : hyperbolic sine function, $\int$ : integral, : parameter, $\beta$ : parameter and $\ell$ : parameter
Permalink:: http://dlmf.nist.gov/20.10.E4
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20.10.5 $\int_{0}^{\infty}e^{{-st}}\mathop{\theta_{{3}}\/}\nolimits\!\left(\frac{(1+% \beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)dt=\int_{0}^{\infty}e^{% {-st}}\mathop{\theta_{{4}}\/}\nolimits\!\left(\frac{\beta\pi}{2\ell}\middle|% \frac{i\pi t}{\ell^{2}}\right)dt=\frac{\ell}{\sqrt{s}}\mathop{\cosh\/}% \nolimits\!\left(\beta\sqrt{s}\right)\mathop{\mathrm{csch}\/}\nolimits\!\left(% \ell\sqrt{s}\right).$

Symbols:: $\mathop{\theta_{{j}}\/}\nolimits\!\left(z\middle|\tau\right)$ : theta function, : differential of , : base of exponential function, $\mathop{\mathrm{csch}\/}\nolimits z$ : hyperbolic cosecant function, $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function, $\int$ : integral, : parameter, $\beta$ : parameter and $\ell$ : parameter
Permalink:: http://dlmf.nist.gov/20.10.E5
Encodings:: TeX, pMML, png

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

Permalink:: http://dlmf.nist.gov/20.10.iii

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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 20.9 Relations to Other Functions 20.11 Generalizations and Analogs

§20.10 Integrals

Contents

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

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

§20.10(iii) Compendia