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 FresnelC

 http://functions.wolfram.com/06.33.21.0129.01

 Input Form

 Integrate[z^(\[Alpha] - 1) FresnelS[b z] FresnelC[a z], z] == (z^\[Alpha]/(2 \[Alpha] (3 + \[Alpha]))) FresnelC[a z] (2 (3 + \[Alpha]) FresnelS[b z] - b^3 Pi z^3 HypergeometricPFQ[ {3/4 + \[Alpha]/4}, {3/2, 7/4 + \[Alpha]/4}, (-(1/16)) b^4 Pi^2 z^4]) - ((2^((1/2) (-2 + \[Alpha])) Pi^(-1 - \[Alpha]/2))/a^3) Sum[(((-1)^k b^(3 + 4 k) z^(4 k + \[Alpha]) (a^4 z^4)^(-2 k - \[Alpha]/2))/ ((3 + 4 k) (3 + 4 k + \[Alpha]) (1 + 2 k)!)) ((I a^2 z^2)^((1/2) (4 k + \[Alpha])) Gamma[\[Alpha]/2 + 2 k + 2, (-(1/2)) I a^2 Pi z^2] + ((-I) a^2 z^2)^((1/2) (4 k + \[Alpha])) Gamma[\[Alpha]/2 + 2 k + 2, (1/2) I a^2 Pi z^2]), {k, 0, Infinity}]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List["FresnelS", "[", RowBox[List["b", " ", "z"]], "]"]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["z", "\[Alpha]"], RowBox[List["2", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["3", "+", "\[Alpha]"]], ")"]]]]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["3", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["FresnelS", "[", RowBox[List["b", " ", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["b", "3"], " ", "\[Pi]", " ", SuperscriptBox["z", "3"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["7", "4"], "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "16"]]], " ", SuperscriptBox["b", "4"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "4"]]]]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Alpha]", "2"]]]]]], SuperscriptBox["a", "3"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["b", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", FractionBox["\[Alpha]", "2"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], " ", "+", RowBox[List["2", "k"]], "+", "2"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], " ", "+", RowBox[List["2", "k"]], "+", "2"]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]]]]]]]]]]]]

 MathML Form

 z α - 1 S TagBox["S", FresnelS] ( b z ) C TagBox["C", FresnelC] ( a z ) z z α C TagBox["C", FresnelC] ( a z ) 2 α ( α + 3 ) ( 2 ( α + 3 ) S TagBox["S", FresnelS] ( b z ) - b 3 π z 3 1 F 2 ( α 4 + 3 4 ; 3 2 , α 4 + 7 4 ; - 1 16 b 4 π 2 z 4 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["3", "4"]]], HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List[FractionBox["\[Alpha]", "4"], "+", FractionBox["7", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "16"]]], " ", SuperscriptBox["b", "4"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "4"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] ) - 2 α - 2 2 π - α 2 - 1 a 3 k = 0 ( - 1 ) k b 4 k + 3 z 4 k + α ( a 4 z 4 ) - 2 k - α 2 ( 4 k + 3 ) ( 4 k + α + 3 ) ( 2 k + 1 ) ! ( ( - a 2 z 2 ) 1 2 ( 4 k + α ) Γ ( 2 k + α 2 + 2 , 1 2 a 2 π z 2 ) + ( a 2 z 2 ) 1 2 ( 4 k + α ) Γ ( 2 k + α 2 + 2 , - 1 2 a 2 π z 2 ) ) z z α -1 FresnelS b z FresnelC a z z α FresnelC a z 2 α α 3 -1 2 α 3 FresnelS b z -1 b 3 z 3 HypergeometricPFQ α 4 -1 3 4 3 2 α 4 -1 7 4 -1 1 16 b 4 2 z 4 -1 2 α -2 2 -1 -1 α 2 -1 -1 a 3 -1 k 0 -1 k b 4 k 3 z 4 k α a 4 z 4 -2 k -1 α 2 -1 4 k 3 4 k α 3 2 k 1 -1 -1 a 2 z 2 1 2 4 k α Gamma 2 k α 2 -1 2 1 2 a 2 z 2 a 2 z 2 1 2 4 k α Gamma 2 k α 2 -1 2 -1 1 2 a 2 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["FresnelS", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["3", "+", "\[Alpha]"]], ")"]], " ", RowBox[List["FresnelS", "[", RowBox[List["b", " ", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["b", "3"], " ", "\[Pi]", " ", SuperscriptBox["z", "3"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Alpha]", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List[FractionBox["7", "4"], "+", FractionBox["\[Alpha]", "4"]]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "16"]]], " ", SuperscriptBox["b", "4"], " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "4"]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", "\[Alpha]", " ", RowBox[List["(", RowBox[List["3", "+", "\[Alpha]"]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Alpha]"]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", FractionBox["\[Alpha]", "2"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["b", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]], ")"]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "-", FractionBox["\[Alpha]", "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", RowBox[List["2", " ", "k"]], "+", "2"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["\[Alpha]", "2"], "+", RowBox[List["2", " ", "k"]], "+", "2"]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["4", " ", "k"]], "+", "\[Alpha]"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]], ")"]], "!"]]]]]]]]], SuperscriptBox["a", "3"]]]]]]]]

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

 2001-10-29