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 FresnelC

 http://functions.wolfram.com/06.33.21.0076.01

 Input Form

 Integrate[z^3 Sinh[b z^2] FresnelC[a z], z] == (I/(16 b^2)) (-8 I FresnelC[a z] (b z^2 Cosh[b z^2] - Sinh[b z^2]) + a ((4 Sqrt[Pi] (-3 I b + a^2 Pi) FresnelS[Sqrt[a^2 - (2 I b)/Pi] z])/ (-2 I b + a^2 Pi)^(3/2) - (4 Sqrt[Pi] (3 I b + a^2 Pi) FresnelS[Sqrt[a^2 + (2 I b)/Pi] z])/(2 I b + a^2 Pi)^(3/2) + ((I b)/z) (Sqrt[2 Pi] (Sqrt[(2 b - I a^2 Pi) z^2]/(2 I b + a^2 Pi)^2 + Sqrt[(-(2 b + I a^2 Pi)) z^2]/(-2 I b + a^2 Pi)^2 - Sqrt[(2 b + I a^2 Pi) z^2]/(2 b + I a^2 Pi)^2 + Sqrt[I (2 I b + a^2 Pi) z^2]/(2 I b + a^2 Pi)^2) + ((8 z^2)/(4 b^2 + a^4 Pi^2)) (a^2 Pi Cosh[b z^2] Sin[(1/2) a^2 Pi z^2] + 2 b Cos[(1/2) a^2 Pi z^2] Sinh[b z^2]))))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List["16", " ", SuperscriptBox["b", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", "\[ImaginaryI]", " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "\[Pi]"]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["4", " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["FresnelS", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "\[Pi]"]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "b"]], "z"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]], "+", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]], "-", FractionBox[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]], "+", FractionBox[SqrtBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], "2"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["8", " ", SuperscriptBox["z", "2"], " "]], RowBox[List[RowBox[List["4", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List["2", " ", "b", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z 3 sinh ( b z 2 ) C TagBox["C", FresnelC] ( a z ) z 16 b 2 ( a ( - 4 π ( π a 2 + 3 b ) S TagBox["S", FresnelS] ( a 2 + 2 b π z ) ( π a 2 + 2 b ) 3 / 2 + 4 π ( a 2 π - 3 b ) S TagBox["S", FresnelS] ( a 2 - 2 b π z ) ( a 2 π - 2 b ) 3 / 2 + b z ( 2 π ( ( π a 2 + 2 b ) z 2 ( π a 2 + 2 b ) 2 + - ( π a 2 + 2 b ) z 2 ( a 2 π - 2 b ) 2 + ( 2 b - a 2 π ) z 2 ( π a 2 + 2 b ) 2 - ( π a 2 + 2 b ) z 2 ( π a 2 + 2 b ) 2 ) + 8 z 2 π 2 a 4 + 4 b 2 ( π cosh ( b z 2 ) sin ( 1 2 a 2 π z 2 ) a 2 + 2 b cos ( 1 2 a 2 π z 2 ) sinh ( b z 2 ) ) ) ) - 8 C TagBox["C", FresnelC] ( a z ) ( b z 2 cosh ( b z 2 ) - sinh ( b z 2 ) ) ) z z 3 b z 2 FresnelC a z 16 b 2 -1 a -1 4 1 2 a 2 3 b FresnelS a 2 2 b -1 1 2 z a 2 2 b 3 2 -1 4 1 2 a 2 -1 3 b FresnelS a 2 -1 2 b -1 1 2 z a 2 -1 2 b 3 2 -1 b z -1 2 1 2 a 2 2 b z 2 1 2 a 2 2 b 2 -1 -1 a 2 2 b z 2 1 2 a 2 -1 2 b 2 -1 2 b -1 a 2 z 2 1 2 a 2 2 b 2 -1 -1 a 2 2 b z 2 1 2 a 2 2 b 2 -1 8 z 2 2 a 4 4 b 2 -1 b z 2 1 2 a 2 z 2 a 2 2 b 1 2 a 2 z 2 b z 2 -1 8 FresnelC a z b z 2 b z 2 -1 b z 2 [/itex]

 Rule Form

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

 2001-10-29