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 FresnelC

 http://functions.wolfram.com/06.33.21.0079.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2001-10-29