Formula:7533

Digital Library of Mathematical Functions ID 20.10.E2

${\displaystyle{\displaystyle\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\mathrm{d}x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)}}$:
• ${\displaystyle{\displaystyle\zeta\left(\NVar{s}\right)}}$: Riemann zeta function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
 

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