Formula:4233

Digital Library of Mathematical Functions ID 10.32.E16

${\displaystyle{\displaystyle I_{\mu}\left(x\right)K_{\nu}\left(x\right)=\int_{0}^{\infty}J_{\mu\pm\nu}\left(2x\sinh t\right)e^{(-\mu\pm\nu)t}\mathrm{d}t,}}$

Constraint(s)

${\displaystyle{\displaystyle\Re(\mu\mp\nu)>-\tfrac{1}{2}}}$
${\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}$
${\displaystyle{\displaystyle\Re(\mu\mp\nu)>-\tfrac{1}{2}}}$
${\displaystyle{\displaystyle\Re(\mu\pm\nu)>-1}}$
${\displaystyle{\displaystyle x>0}}$

Symbols List

• ${\displaystyle{\displaystyle J_{\NVar{\nu}}\left(\NVar{z}\right)}}$: Bessel function of the first kind
• ${\displaystyle{\displaystyle\mathrm{d}\NVar{x}}}$: differential of x
• ${\displaystyle{\displaystyle\mathrm{e}}}$: base of natural logarithm
• ${\displaystyle{\displaystyle\sinh\NVar{z}}}$: hyperbolic sine function
• ${\displaystyle{\displaystyle\int}}$: integral
• ${\displaystyle{\displaystyle I_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the first kind
• ${\displaystyle{\displaystyle K_{\NVar{\nu}}\left(\NVar{z}\right)}}$: modified Bessel function of the second kind
• ${\displaystyle{\displaystyle\Re}}$: real part
• ${\displaystyle{\displaystyle x}}$: real variable
• ${\displaystyle{\displaystyle\nu}}$: complex parameter
 

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