html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.0591.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18