Formula:4925

Digital Library of Mathematical Functions ID 12.12.E3

${\displaystyle{\displaystyle\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}U\left(a,t\right)\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(\tfrac{1}{2}-a\right)x^{-a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right),}}$

Constraint(s)

${\displaystyle{\displaystyle\Re a<\tfrac{1}{2},x>0}}$

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
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle U\left(\NVar{a},\NVar{z}\right)}}$:
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle a}}$: real or complex parameter
 

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