Fresnel integral C: Integration (formula 06.33.21.0135)

FresnelC

$Mathematica Notation$

Involving the direct function

http://functions.wolfram.com/06.33.21.0135.01

Input Form

Integrate[(t^(\[Alpha] - 1) FresnelC[t])/E^(z t), {t, 0, Infinity}] == -((1/(\[Alpha] (1 + \[Alpha]))) ((Cos[(Pi/4) (1 + \[Alpha])] Gamma[(3 + \[Alpha])/2] HypergeometricPFQ[{(\[Alpha] + 1)/4, (\[Alpha] + 3)/4, \[Alpha]/4}, {1/4, 1/2, 3/4, 1 + \[Alpha]/4}, -(z^4/(16 Pi^2))])/(Pi/2)^((\[Alpha] + 1)/2))) + (1/6) ((3 Gamma[\[Alpha]])/z^\[Alpha] - (1/(2 + \[Alpha])) ((3 2^((1 + \[Alpha])/2) z^2 Cos[(1/4) Pi (3 + \[Alpha])] Gamma[(3 + \[Alpha])/2] HypergeometricPFQ[{(\[Alpha] + 2)/4, (\[Alpha] + 3)/4, (\[Alpha] + 5)/4}, {3/4, 5/4, 3/2, (\[Alpha] + 6)/4}, -(z^4/(16 Pi^2))])/Pi^((\[Alpha] + 3)/2)) - (1/(3 + \[Alpha])) (2^((2 + \[Alpha])/2) Pi^(-2 - \[Alpha]/2) z^3 Cos[(Pi \[Alpha])/4] Gamma[2 + \[Alpha]/2] HypergeometricPFQ[ {(\[Alpha] + 3)/4, 1 + \[Alpha]/4, (\[Alpha] + 6)/4}, {5/4, 3/2, 7/4, (\[Alpha] + 7)/4}, -(z^4/(16 Pi^2))]) - (1/(1 + \[Alpha])) (3 (Pi/2)^(-1 - \[Alpha]/2) z Gamma[1 + \[Alpha]/2] HypergeometricPFQ[{(\[Alpha] + 1)/4, (\[Alpha] + 2)/4, 1 + \[Alpha]/4}, {1/2, 3/4, 5/4, (\[Alpha] + 5)/4}, -(z^4/(16 Pi^2))] Sin[(Pi \[Alpha])/4])) /; Re[z] > 0 && Re[\[Alpha]] > -1

Standard Form

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MathML Form

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

2001-10-29