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http://functions.wolfram.com/06.32.21.0017.01
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Integrate[(t^(\[Alpha] - 1) FresnelS[t])/E^(z t), {t, 0, Infinity}] ==
((1/2) Gamma[\[Alpha]])/z^\[Alpha] + (1/(1 + \[Alpha])) 2^(\[Alpha]/2)
Pi^(-1 - \[Alpha]/2) z Cos[(Pi \[Alpha])/4] 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))] +
((1/(2 \[Alpha])) Cos[(Pi/4) (3 + \[Alpha])] Gamma[(1 + \[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/(2 + \[Alpha])) 2^((\[Alpha] - 1)/2) z^2
Cos[(Pi/4) (5 + \[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 (3 + \[Alpha]))) 2^(\[Alpha]/2) Pi^(-2 - \[Alpha]/2) z^3
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))] Sin[(Pi \[Alpha])/4] /; Re[z] > 0 && Re[\[Alpha]] > -3
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