Airy function Ai: Integral transforms (formula 03.05.22.0008)

AiryAi

$Mathematica Notation$

Bessel-Type Functions

AiryAi[z]

Integral transforms

Hankel transforms

http://functions.wolfram.com/03.05.22.0008.01

Input Form

HankelTransform[AiryAi[t], {t, \[Nu]}, z] == 2^(-6 - \[Nu]) 3^((1/3) (-1 - \[Nu])) z^(1/2 + \[Nu]) ((1/(Pi Gamma[1 + \[Nu]])) (32 3^(1/6 + \[Nu]) Gamma[1/2 + \[Nu]/3] Gamma[5/6 + \[Nu]/3] HypergeometricPFQ[{1/4 + \[Nu]/6, 5/12 + \[Nu]/6, 3/4 + \[Nu]/6, 11/12 + \[Nu]/6}, {1/3, 2/3, 1/3 + \[Nu]/3, 2/3 + \[Nu]/3, 1 + \[Nu]/3}, -(z^6/36)]) - (16 z^2 Gamma[7/6 + \[Nu]/3] Gamma[3/2 + \[Nu]/3] HypergeometricPFQ[{7/12 + \[Nu]/6, 3/4 + \[Nu]/6, 13/12 + \[Nu]/6, 5/4 + \[Nu]/6}, {2/3, 4/3, 2/3 + \[Nu]/3, 1 + \[Nu]/3, 4/3 + \[Nu]/3}, -(z^6/36)])/(Gamma[1 + \[Nu]/3] Gamma[(2 + \[Nu])/3] Gamma[(4 + \[Nu])/3]) + (1/(Pi Gamma[3 + \[Nu]])) (3^(17/6 + \[Nu]) z^4 Gamma[11/6 + \[Nu]/3] Gamma[13/6 + \[Nu]/3] HypergeometricPFQ[{11/12 + \[Nu]/6, 13/12 + \[Nu]/6, 17/12 + \[Nu]/6, 19/12 + \[Nu]/6}, {4/3, 5/3, 1 + \[Nu]/3, 4/3 + \[Nu]/3, 5/3 + \[Nu]/3}, -(z^6/36)])) /; z > 0 && Re[\[Nu]] > -(3/2)

Standard Form

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MathML Form

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

2001-10-29