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 FresnelC

 http://functions.wolfram.com/06.33.21.0084.01

 Input Form

 Integrate[z^3 Cosh[b z^2] FresnelC[a z], z] == (-(1/(32 b^2))) (2 I a Sqrt[2 Pi] (((-3 I b + a^2 Pi) Erfi[Sqrt[-b - (1/2) I a^2 Pi] z])/ (-2 b - I a^2 Pi)^(3/2) + ((3 I b + a^2 Pi) Erfi[Sqrt[b - (1/2) I a^2 Pi] z])/(2 b - I a^2 Pi)^(3/2) - ((3 I b + a^2 Pi) Erfi[Sqrt[-b + (1/2) I a^2 Pi] z])/ (-2 b + I a^2 Pi)^(3/2) + ((3 I b - a^2 Pi) Erfi[Sqrt[b + (1/2) I a^2 Pi] z])/(2 b + I a^2 Pi)^(3/2)) + 16 FresnelC[a z] (Cosh[b z^2] - b z^2 Sinh[b z^2]) + ((16 a b z)/(4 b^2 + a^4 Pi^2)) (2 b Cos[(1/2) a^2 Pi z^2] Cosh[b z^2] + a^2 Pi Sin[(1/2) a^2 Pi z^2] Sinh[b z^2]))

 Standard Form

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 MathML Form

 z 3 cosh ( b z 2 ) C TagBox["C", FresnelC] ( a z ) z - 1 32 b 2 ( 2 a 2 π ( - ( π a 2 + 3 b ) erfi ( 1 2 a 2 π - b z ) ( a 2 π - 2 b ) 3 / 2 + ( - π a 2 + 3 b ) erfi ( 1 2 π a 2 + b z ) ( π a 2 + 2 b ) 3 / 2 + ( a 2 π - 3 b ) erfi ( - 1 2 π a 2 - b z ) ( - π a 2 - 2 b ) 3 / 2 + ( π a 2 + 3 b ) erfi ( b - 1 2 a 2 π z ) ( 2 b - a 2 π ) 3 / 2 ) + 16 a b z π 2 a 4 + 4 b 2 ( π sin ( 1 2 a 2 π z 2 ) sinh ( b z 2 ) a 2 + 2 b cos ( 1 2 a 2 π z 2 ) cosh ( b z 2 ) ) + 16 C TagBox["C", FresnelC] ( a z ) ( cosh ( b z 2 ) - b z 2 sinh ( b z 2 ) ) ) z z 3 b z 2 FresnelC a z -1 1 32 b 2 -1 2 a 2 1 2 -1 a 2 3 b Erfi 1 2 a 2 -1 b 1 2 z a 2 -1 2 b 3 2 -1 -1 a 2 3 b Erfi 1 2 a 2 b 1 2 z a 2 2 b 3 2 -1 a 2 -1 3 b Erfi -1 1 2 a 2 -1 b 1 2 z -1 a 2 -1 2 b 3 2 -1 a 2 3 b Erfi b -1 1 2 a 2 1 2 z 2 b -1 a 2 3 2 -1 16 a b z 2 a 4 4 b 2 -1 1 2 a 2 z 2 b z 2 a 2 2 b 1 2 a 2 z 2 b z 2 16 FresnelC a z b z 2 -1 b z 2 b z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", RowBox[List["Cosh", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[ImaginaryI]", " ", "b"]], "-", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]], "+", RowBox[List["16", " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], "-", RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["16", " ", "a", " ", "b", " ", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List["4", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], RowBox[List["32", " ", SuperscriptBox["b", "2"]]]]]]]]]]

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

 2001-10-29