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 FresnelS

 http://functions.wolfram.com/06.32.21.0011.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]]]], SqrtBox["z"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List["-", RowBox[List[FractionBox["1", "b"], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[ImaginaryI]", "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]]]], "-", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]], "-", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["4", "+", RowBox[List["4", " ", "\[ImaginaryI]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]]]

 MathML Form

 cos ( b z ) S TagBox["S", FresnelS] ( a z ) z z - 1 b ( ( 1 4 - 4 ) - b 2 2 a 2 π ( b 2 a 2 π erf ( ( 1 2 + 2 ) ( a 2 π z - b ) a π ) - b 2 a 2 π erf ( ( 1 2 + 2 ) ( π z a 2 + b ) a π ) - erfi ( ( 1 2 + 2 ) ( b - a 2 π z ) a π ) - erfi ( ( 1 2 + 2 ) ( π z a 2 + b ) a π ) - ( 4 + 4 ) b 2 2 a 2 π S TagBox["S", FresnelS] ( a z ) sin ( b z ) ) ) z b z 1 2 FresnelS a z 1 2 z 1 2 -1 -1 1 b -1 1 4 -1 4 -1 -1 b 2 2 a 2 -1 b 2 a 2 -1 Erf 1 2 2 -1 a 2 z 1 2 -1 b a 1 2 -1 -1 b 2 a 2 -1 Erf 1 2 2 -1 z 1 2 a 2 b a 1 2 -1 -1 Erfi 1 2 2 -1 b -1 a 2 z 1 2 a 1 2 -1 -1 Erfi 1 2 2 -1 z 1 2 a 2 b a 1 2 -1 -1 4 4 b 2 2 a 2 -1 FresnelS a z 1 2 b z 1 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["FresnelS", "[", RowBox[List["a_", " ", SqrtBox["z_"]]], "]"]]]], SqrtBox["z_"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[ImaginaryI]", "4"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]]]], "-", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "-", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]], "-", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[ImaginaryI]", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List[SuperscriptBox["a", "2"], " ", "\[Pi]", " ", SqrtBox["z"]]]]], ")"]]]], RowBox[List["a", " ", SqrtBox["\[Pi]"]]]], "]"]], "-", RowBox[List[RowBox[List["(", RowBox[List["4", "+", RowBox[List["4", " ", "\[ImaginaryI]"]]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["b", "2"]]], RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", "\[Pi]"]]]], " ", RowBox[List["FresnelS", "[", RowBox[List["a", " ", SqrtBox["z"]]], "]"]], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]]]], ")"]]]], "b"]]]]]]]

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

 2001-10-29

© 1998-2013 Wolfram Research, Inc.