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/01.20.21.0583.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Sin", "[", RowBox[List[RowBox[List["b_", " ", SqrtBox["z_"]]], "+", RowBox[List["d_", " ", "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[RowBox[List["-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", 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["Cos", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]], "]"]]]]]], ")"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cos", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]], "]"]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sin", "[", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["\[ImaginaryI]", " ", "f"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]]]]

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

 2002-12-18