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 FresnelC

 http://functions.wolfram.com/06.33.21.0021.01

 Input Form

 Integrate[z^3 E^(b z^2) FresnelC[a z], z] == (1/(8 b^2)) ((-((4 a b z)/(4 b^2 + a^4 Pi^2))) (2 b Cos[(1/2) a^2 Pi z^2] + a^2 Pi Sin[(1/2) a^2 Pi z^2]) Exp[b z^2] + ((Sqrt[Pi] a (3 b - I a^2 Pi))/(2 (b - (1/2) I a^2 Pi)^(3/2))) Erfi[Sqrt[b - (1/2) I a^2 Pi] z] + ((Sqrt[Pi] a (3 b + I a^2 Pi))/(2 (b + (1/2) I a^2 Pi)^(3/2))) Erfi[Sqrt[b + (1/2) I a^2 Pi] z] + 4 E^(b z^2) (-1 + b z^2) FresnelC[a z])

 Standard Form

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

 MathML Form

 z 3 b z 2 C TagBox["C", FresnelC] ( a z ) z 1 8 b 2 ( π a ( π a 2 + 3 b ) erfi ( 1 2 π a 2 + b z ) 2 ( 1 2 π a 2 + b ) 3 / 2 + π a ( 3 b - a 2 π ) erfi ( b - 1 2 a 2 π z ) 2 ( b - 1 2 a 2 π ) 3 / 2 + 4 b z 2 ( b z 2 - 1 ) C TagBox["C", FresnelC] ( a z ) - 4 a b z b z 2 π 2 a 4 + 4 b 2 ( π sin ( 1 2 a 2 π z 2 ) a 2 + 2 b cos ( 1 2 a 2 π z 2 ) ) ) z z 3 b z 2 FresnelC a z 1 8 b 2 -1 1 2 a a 2 3 b Erfi 1 2 a 2 b 1 2 z 2 1 2 a 2 b 3 2 -1 1 2 a 3 b -1 a 2 Erfi b -1 1 2 a 2 1 2 z 2 b -1 1 2 a 2 3 2 -1 4 b z 2 b z 2 -1 FresnelC a z -1 4 a b z b z 2 2 a 4 4 b 2 -1 1 2 a 2 z 2 a 2 2 b 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_", "3"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["FresnelC", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "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[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SuperscriptBox["z", "2"]]], "]"]]]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]]]], RowBox[List[RowBox[List["4", " ", SuperscriptBox["b", "2"]]], "+", RowBox[List[SuperscriptBox["a", "4"], " ", SuperscriptBox["\[Pi]", "2"]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", "a", " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]]]], ")"]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]], " ", "z"]], "]"]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "+", RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["FresnelC", "[", RowBox[List["a", " ", "z"]], "]"]]]]]], RowBox[List["8", " ", SuperscriptBox["b", "2"]]]]]]]]

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

 2001-10-29