Formula:4115

Digital Library of Mathematical Functions ID 10.22.E44

${\displaystyle{\displaystyle\int_{0}^{\infty}t^{\mu}Y_{\nu}\left(t\right)\mathrm{d}t=\frac{2^{\mu}}{\pi}\Gamma\left(\tfrac{1}{2}\mu+\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\Gamma\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu+\tfrac{1}{2}\right)\sin\left(\tfrac{1}{2}\mu-\tfrac{1}{2}\nu\right)\pi,}}$

Constraint(s)

${\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}$
${\displaystyle{\displaystyle\Re\mu<\tfrac{1}{2}}}$

Symbols List

• ${\displaystyle{\displaystyle Y_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the second kind
• ${\displaystyle{\displaystyle\Gamma\left(\NVar{z}\right)}}$: 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
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle\sin\NVar{z}}}$: sine function
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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