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 FresnelS

 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[SuperscriptBox["t", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z"]], " ", "t"]]], " ", RowBox[List["FresnelS", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["1", "+", "\[Alpha]"]]], SuperscriptBox["2", RowBox[List["\[Alpha]", "/", "2"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Alpha]", "2"]]]], " ", "z", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "4"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "5"]], "4"]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", "\[Alpha]"]]], SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", "2"], ")"]], RowBox[List["-", FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "2"]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["\[Pi]", "4"], RowBox[List["(", RowBox[List["3", "+", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], ",", FractionBox["\[Alpha]", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "4"], ",", FractionBox["1", "2"], ",", FractionBox["3", "4"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", "+", "\[Alpha]"]]], SuperscriptBox["2", FractionBox[RowBox[List["\[Alpha]", "-", "1"]], "2"]], " ", SuperscriptBox["\[Pi]", RowBox[List["-", FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "2"]]]], " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["\[Pi]", "4"], RowBox[List["(", RowBox[List["5", "+", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "5"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "6"]], "4"]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], "]"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["3", " ", RowBox[List["(", RowBox[List["3", "+", "\[Alpha]"]], ")"]]]]], SuperscriptBox["2", RowBox[List["\[Alpha]", "/", "2"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "2"]], "-", FractionBox["\[Alpha]", "2"]]]], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", FractionBox["\[Alpha]", "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]], ",", FractionBox[RowBox[List["\[Alpha]", "+", "6"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "4"], ",", FractionBox["3", "2"], ",", FractionBox["7", "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "7"]], "4"]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", RowBox[List["-", "3"]]]]]]]]]]

 MathML Form

 0 t α - 1 - z t S TagBox["S", FresnelS] ( t ) t 1 2 z - α Γ ( α ) + 2 α / 2 π - α 2 - 1 z α + 1 cos ( π α 4 ) Γ ( α 2 + 1 ) 3 F 4 ( α + 1 4 , α + 2 4 , α 4 + 1 ; 1 2 , 3 4 , 5 4 , α + 5 4 ; - z 4 16 π 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "5"]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 2 α ( π 2 ) - α + 1 2 cos ( π 4 ( α + 3 ) ) Γ ( α + 1 2 ) 3 F 4 ( α + 1 4 , α + 3 4 , α 4 ; 1 4 , 1 2 , 3 4 , α 4 + 1 ; - z 4 16 π 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "1"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["\[Alpha]", "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["1", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] + 1 α + 2 2 α - 1 2 π - α + 3 2 z 2 cos ( π 4 ( α + 5 ) ) Γ ( α + 3 2 ) 3 F 4 ( α + 2 4 , α + 3 4 , α + 5 4 ; 3 4 , 5 4 , 3 2 , α + 6 4 ; - z 4 16 π 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "5"]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "6"]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] - 2 α / 2 π - α 2 - 2 z 3 3 ( α + 3 ) Γ ( α 2 + 2 ) sin ( π α 4 ) 3 F 4 ( α + 3 4 , α 4 + 1 , α + 6 4 ; 5 4 , 3 2 , 7 4 , α + 7 4 ; - z 4 16 π 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["4", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", "1"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "6"]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["7", "4"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["\[Alpha]", "+", "7"]], "4"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] /; Re ( z ) > 0 Re ( α ) > - 3 Condition t 0 t α -1 -1 z t FresnelS t 1 2 z -1 α Gamma α 2 α 2 -1 -1 α 2 -1 -1 z α 1 -1 cos α 4 -1 Gamma α 2 -1 1 HypergeometricPFQ α 1 4 -1 α 2 4 -1 α 4 -1 1 1 2 3 4 5 4 α 5 4 -1 -1 z 4 16 2 -1 1 2 α -1 2 -1 -1 α 1 2 -1 4 -1 α 3 Gamma α 1 2 -1 HypergeometricPFQ α 1 4 -1 α 3 4 -1 α 4 -1 1 4 1 2 3 4 α 4 -1 1 -1 z 4 16 2 -1 1 α 2 -1 2 α -1 2 -1 -1 α 3 2 -1 z 2 4 -1 α 5 Gamma α 3 2 -1 HypergeometricPFQ α 2 4 -1 α 3 4 -1 α 5 4 -1 3 4 5 4 3 2 α 6 4 -1 -1 z 4 16 2 -1 -1 2 α 2 -1 -1 α 2 -1 -2 z 3 3 α 3 -1 Γ α 2 -1 2 α 4 -1 HypergeometricPFQ α 3 4 -1 α 4 -1 1 α 6 4 -1 5 4 3 2 7 4 α 7 4 -1 -1 z 4 16 2 -1 z 0 α -3 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "z_"]], " ", "t_"]]], " ", RowBox[List["FresnelS", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Gamma", "[", "\[Alpha]", "]"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "/", "2"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Alpha]", "2"]]]], " ", "z", " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", FractionBox["\[Alpha]", "2"]]], "]"]], " ", 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RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["5", "+", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "2"]], "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "5"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "4"], ",", FractionBox["5", "4"], ",", FractionBox["3", "2"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "6"]], "4"]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], "]"]]]], RowBox[List["2", "+", "\[Alpha]"]]], "-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Alpha]", "/", "2"]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "2"]], "-", FractionBox["\[Alpha]", "2"]]]], " ", SuperscriptBox["z", "3"], " ", RowBox[List["Gamma", "[", RowBox[List["2", "+", FractionBox["\[Alpha]", "2"]]], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Alpha]", "+", "3"]], "4"], ",", RowBox[List["1", "+", FractionBox["\[Alpha]", "4"]]], ",", FractionBox[RowBox[List["\[Alpha]", "+", "6"]], "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "4"], ",", FractionBox["3", "2"], ",", FractionBox["7", "4"], ",", FractionBox[RowBox[List["\[Alpha]", "+", "7"]], "4"]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "4"], RowBox[List["16", " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Alpha]"]], "4"], "]"]]]], RowBox[List["3", " ", RowBox[List["(", RowBox[List["3", "+", "\[Alpha]"]], ")"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Alpha]", "]"]], ">", RowBox[List["-", "3"]]]]]]]]]]]]

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

 2001-10-29