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 FresnelC

 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

 Cell[BoxData[RowBox[List[RowBox[List[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["FresnelC", "[", "t", "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["\[Alpha]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]]], RowBox[List["(", RowBox[List[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["1", "+", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", 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 MathML Form

 0 t α - 1 - z t C TagBox["C", FresnelC] ( t ) t 1 6 ( 3 z - α Γ ( α ) - 3 z 2 α + 2 2 α + 1 2 π - α + 3 2 cos ( π 4 ( α + 3 ) ) 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] - z 3 α + 3 2 α + 2 2 π - α 2 - 2 cos ( π α 4 ) Γ ( α 2 + 2 ) 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] - 3 z α + 1 ( π 2 ) - α 2 - 1 Γ ( α 2 + 1 ) sin ( π α 4 ) 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 α ( α + 1 ) ( π 2 ) - α + 1 2 cos ( π 4 ( α + 1 ) ) Γ ( α + 3 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] /; Re ( z ) > 0 Re ( α ) > - 1 Condition t 0 t α -1 -1 z t FresnelC t 1 6 3 z -1 α Gamma α -1 3 z 2 α 2 -1 2 α 1 2 -1 -1 α 3 2 -1 4 -1 α 3 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 z 3 α 3 -1 2 α 2 2 -1 -1 α 2 -1 -2 α 4 -1 Gamma α 2 -1 2 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 -1 3 z α 1 -1 2 -1 -1 α 2 -1 -1 Gamma α 2 -1 1 α 4 -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 1 α α 1 -1 2 -1 -1 α 1 2 -1 4 -1 α 1 Gamma α 3 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 z 0 α -1 [/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["FresnelC", "[", "t_", "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["\[Pi]", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "+", "1"]], ")"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Alpha]"]], ")"]]]], "]"]], " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["3", "+", "\[Alpha]"]], "2"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", 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 Date Added to functions.wolfram.com (modification date)

 2001-10-29