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 Cosh

 http://functions.wolfram.com/01.20.21.0747.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", RowBox[List["f_", " ", "z_"]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cosh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["\[ImaginaryI]", " ", "f", " ", "z"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cosh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["\[ImaginaryI]", " ", "f", " ", "z"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

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

 2002-12-18