Formula:4926

Digital Library of Mathematical Functions ID 12.12.E4

${\displaystyle{\displaystyle(U\left(a,z\right))^{2}+(\overline{U}\left(a,z\right))^{2}=\frac{2^{\frac{3}{2}}}{\pi}\Gamma\left(\tfrac{1}{2}-a\right)\int_{0}^{\infty}\frac{e^{2at+\frac{1}{2}z^{2}\tanh t}}{\sqrt{\sinh\left(2t\right)}}\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle\Re a<\tfrac{1}{2}}}$

Symbols List

• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: gamma function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\tanh\NVar{z}}}$: hyperbolic tangent function
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle\overline{U}\left(\NVar{a},\NVar{x}\right)}}$:
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
 

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