Modified Bessel function of the second kind: Integration (formula 03.04.21.0031)

BesselK

$Mathematica Notation$

Bessel-Type Functions

BesselK[nu,z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving hyperbolic functions

Involving sinh

Power arguments

http://functions.wolfram.com/03.04.21.0031.01

Input Form

Integrate[Sinh[a z^r] BesselK[\[Nu], a z^r], z] == -(2^(-1 - \[Nu]) z (a z^r)^(1 - \[Nu]) (4^\[Nu] ((1 + r) Gamma[\[Nu]] + r Gamma[1 + \[Nu]]) HypergeometricPFQ[{3/4 - \[Nu]/2, 5/4 - \[Nu]/2, 1/2 + 1/(2 r) - \[Nu]/2}, {3/2, 1 - \[Nu], 3/2 - \[Nu], 3/2 + 1/(2 r) - \[Nu]/2}, a^2 z^(2 r)] + (a z^r)^(2 \[Nu]) (r Gamma[1 - \[Nu]] + (1 + r) Gamma[-\[Nu]]) HypergeometricPFQ[ {3/4 + \[Nu]/2, 5/4 + \[Nu]/2, 1/2 + 1/(2 r) + \[Nu]/2}, {3/2, 3/2 + 1/(2 r) + \[Nu]/2, 1 + \[Nu], 3/2 + \[Nu]}, a^2 z^(2 r)]))/ ((-1 + r (-1 + \[Nu])) (1 + r + r \[Nu]))

Standard Form

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MathML Form

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<apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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Date Added to functions.wolfram.com (modification date)

2001-10-29