Formula:7535

Digital Library of Mathematical Functions ID 20.10.E4

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "TeX input is invalid."): {\displaystyle \int_{0}^{\infty}e^{-st}\Jacobithetatau{1}@{\frac{\beta\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}=\int_{0}^{\infty}e^{-st}\Jacobithetatau{2}@{\frac{(1+\beta)\pi}{2\ell}}{\frac{i\pi t}{\ell^{2}}}\diff{t}=-\frac{\ell}{\sqrt{s}}\sinh@{\beta\sqrt{s}}\sech@{\ell\sqrt{s}},}

Symbols List

• ${\displaystyle{\displaystyle\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)}}$:
• Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "TeX input is invalid."): {\displaystyle \cpi} : the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\operatorname{sech}\NVar{z}}}$: hyperbolic secant function
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\mathrm{i}}}$: imaginary unit
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle s}}$: parameter
• ${\displaystyle{\displaystyle\beta}}$: parameter
• ${\displaystyle{\displaystyle\ell}}$: parameter
 

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