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 FresnelC

 http://functions.wolfram.com/06.33.21.0105.01

 Input Form

 Integrate[z Log[b z] FresnelC[a z], z] == (1/(144 a^5 Pi z^3)) (9 (I Sqrt[2] Sqrt[a^4 z^4] (Sqrt[(-I) a^2 z^2] - Sqrt[I a^2 z^2]) + 4 a^4 z^4 Sin[(1/2) a^2 Pi z^2]) + 4 a^6 Pi z^6 HypergeometricPFQ[{3/2, 3/2}, {5/2, 5/2}, (-(1/2)) I a^2 Pi z^2] + 4 a^6 Pi z^6 HypergeometricPFQ[{3/2, 3/2}, {5/2, 5/2}, (1/2) I a^2 Pi z^2] + 18 I Sqrt[2] Sqrt[a^4 z^4] (Sqrt[(-I) a^2 z^2] - Sqrt[I a^2 z^2]) Log[z] - 18 I Log[b z] (Sqrt[2] Sqrt[a^4 z^4] (Sqrt[(-I) a^2 z^2] - Sqrt[I a^2 z^2]) - 4 I a^4 z^4 Sin[(1/2) a^2 Pi z^2]) + 36 a^5 Pi z^5 FresnelC[a z] (-1 + 2 Log[b z]) + 36 a^3 z^3 FresnelS[a z] (-1 + 2 Log[b z]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List["z", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["144", " ", SuperscriptBox["a", "5"], " ", "\[Pi]", " ", SuperscriptBox["z", "3"]]]], RowBox[List["(", RowBox[List[RowBox[List["9", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], "-", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "6"], " ", "\[Pi]", " ", SuperscriptBox["z", "6"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], ",", FractionBox["5", "2"]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "6"], " ", "\[Pi]", " ", SuperscriptBox["z", "6"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], ",", FractionBox["5", "2"]]], "}"]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List["18", " ", "\[ImaginaryI]", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], "-", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["18", " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], "-", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], ")"]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["36", " ", SuperscriptBox["a", "5"], " ", "\[Pi]", " ", SuperscriptBox["z", "5"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["36", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z log ( b z ) C TagBox["C", FresnelC] ( a z ) z 1 144 a 5 π z 3 ( 4 a 6 π 2 F 2 ( 3 2 , 3 2 ; 5 2 , 5 2 ; 1 2 a 2 π z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] z 6 + 4 a 6 π 2 F 2 ( 3 2 , 3 2 ; 5 2 , 5 2 ; - 1 2 a 2 π z 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["2", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["3", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[FractionBox["5", "2"], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[FractionBox["5", "2"], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] z 6 + 36 a 5 π C TagBox["C", FresnelC] ( a z ) ( 2 log ( b z ) - 1 ) z 5 + 36 a 3 S TagBox["S", FresnelS] ( a z ) ( 2 log ( b z ) - 1 ) z 3 + 18 2 a 4 z 4 ( - a 2 z 2 - a 2 z 2 ) log ( z ) + 9 ( 4 a 4 sin ( 1 2 a 2 π z 2 ) z 4 + 2 a 4 z 4 ( - a 2 z 2 - a 2 z 2 ) ) - 18 log ( b z ) ( 2 a 4 z 4 ( - a 2 z 2 - a 2 z 2 ) - 4 a 4 z 4 sin ( 1 2 a 2 π z 2 ) ) ) z z b z FresnelC a z 1 144 a 5 z 3 -1 4 a 6 HypergeometricPFQ 3 2 3 2 5 2 5 2 1 2 a 2 z 2 z 6 4 a 6 HypergeometricPFQ 3 2 3 2 5 2 5 2 -1 1 2 a 2 z 2 z 6 36 a 5 FresnelC a z 2 b z -1 z 5 36 a 3 FresnelS a z 2 b z -1 z 3 18 2 1 2 a 4 z 4 1 2 -1 a 2 z 2 1 2 -1 a 2 z 2 1 2 z 9 4 a 4 1 2 a 2 z 2 z 4 2 1 2 a 4 z 4 1 2 -1 a 2 z 2 1 2 -1 a 2 z 2 1 2 -1 18 b z 2 1 2 a 4 z 4 1 2 -1 a 2 z 2 1 2 -1 a 2 z 2 1 2 -1 4 a 4 z 4 1 2 a 2 z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["z_", " ", RowBox[List["Log", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["9", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], "-", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "6"], " ", "\[Pi]", " ", SuperscriptBox["z", "6"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], ",", FractionBox["5", "2"]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List["4", " ", SuperscriptBox["a", "6"], " ", "\[Pi]", " ", SuperscriptBox["z", "6"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", FractionBox["3", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["5", "2"], ",", FractionBox["5", "2"]]], "}"]], ",", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]]]], "]"]]]], "+", RowBox[List["18", " ", "\[ImaginaryI]", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], "-", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["18", " ", "\[ImaginaryI]", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"]]]], " ", RowBox[List["(", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]], "-", SqrtBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]]]]], ")"]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "4"], " ", SuperscriptBox["z", "4"], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], "+", RowBox[List["36", " ", SuperscriptBox["a", "5"], " ", "\[Pi]", " ", SuperscriptBox["z", "5"], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]], "+", RowBox[List["36", " ", SuperscriptBox["a", "3"], " ", SuperscriptBox["z", "3"], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["b", " ", "z"]], "]"]]]]]], ")"]]]]]], RowBox[List["144", " ", SuperscriptBox["a", "5"], " ", "\[Pi]", " ", SuperscriptBox["z", "3"]]]]]]]]

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

 2001-10-29