Hermite function: Integration (formula 07.01.21.0009)

HermiteH

$Mathematica Notation$

Hypergeometric Functions

HermiteH[nu,z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Involving exponential function and a power function

http://functions.wolfram.com/07.01.21.0009.01

Input Form

Integrate[(z^(\[Alpha] - 1) HermiteH[\[Nu], a z])/E^(a^2 z^2), z] == 2^(-1 + \[Nu]) Pi z^\[Alpha] ((1/Gamma[(1 - \[Nu])/2]) (Gamma[\[Alpha]/2] HypergeometricPFQRegularized[ {(1 + \[Nu])/2, \[Alpha]/2}, {1/2, (2 + \[Alpha])/2}, (-a^2) z^2]) - (1/Gamma[-(\[Nu]/2)]) (a z Gamma[(1 + \[Alpha])/2] HypergeometricPFQRegularized[{(2 + \[Nu])/2, (1 + \[Alpha])/2}, {3/2, (3 + \[Alpha])/2}, (-a^2) z^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

HermiteH[n, z]