Formula:4228

Digital Library of Mathematical Functions ID 10.32.E11

${\displaystyle{\displaystyle K_{\nu}\left(xz\right)=\frac{\Gamma\left(\nu+\frac{1}{2}\right)(2z)^{\nu}}{\pi^{\frac{1}{2}}x^{\nu}}\int_{0}^{\infty}\frac{\cos\left(xt\right)\mathrm{d}t}{(t^{2}+z^{2})^{\nu+\frac{1}{2}}},}}$

Constraint(s)

${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}$
${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}$
${\displaystyle{\displaystyle\Re\nu>-\tfrac{1}{2}}}$
${\displaystyle{\displaystyle x>0}}$
${\displaystyle{\displaystyle|\operatorname{ph}z|<\tfrac{1}{2}\pi}}$

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
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the second kind
• ${\displaystyle{\displaystyle\operatorname{ph}}}$: phase
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle z}}$: complex variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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