Modified Bessel function of the first kind: Integration (formula 03.02.21.0087)

BesselI

$Mathematica Notation$

Bessel-Type Functions

BesselI[nu,z]

Integration

Definite integration

For the direct function itself

http://functions.wolfram.com/03.02.21.0087.01

Input Form

Integrate[(t^(\[Alpha] - 1) BesselI[\[Nu], t])/E^t, {t, 0, Infinity}] == (1/2) ((2^(\[Alpha] - 1) Cos[(1/2) Pi (\[Alpha] + \[Nu])] Gamma[(\[Alpha] + \[Nu])/2] Hypergeometric2F1[(\[Alpha] - \[Nu])/2, (\[Alpha] + \[Nu])/2, 1/2, 1])/Gamma[(1/2) (2 - \[Alpha] + \[Nu])] + (Gamma[\[Alpha] + \[Nu]] Hypergeometric2F1[(\[Alpha] + \[Nu])/2, (1/2) (1 + \[Alpha] + \[Nu]), 1 + \[Nu], 1])/ (2^\[Nu] Gamma[1 + \[Nu]]) + 2^\[Alpha] Gamma[(1/2) (1 + \[Alpha] + \[Nu])] Hypergeometric2F1[ (1/2) (1 + \[Alpha] - \[Nu]), (1/2) (1 + \[Alpha] + \[Nu]), 3/2, 1] (Sin[(1/2) Pi (\[Alpha] + \[Nu])]/ Gamma[(1/2) (1 - \[Alpha] + \[Nu])])) /; Re[\[Alpha] + \[Nu]] > 0 && Re[\[Alpha]] < 1/2

Standard Form

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MathML Form

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Rule Form

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

2001-10-29