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FresnelS






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > FresnelS[z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/06.32.21.0017.01









  


  










Input Form





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










Standard Form





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MathML Form







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





2001-10-29





© 1998- Wolfram Research, Inc.