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/01.20.21.0839.01

 Input Form

 Integrate[Cos[b Sqrt[z]]^m Cosh[c Sqrt[z] + f z + g], z] == (1/(I f)^(3/2)) (2^(-1 - m) Binomial[m, m/2] (1 - Mod[m, 2]) (c Sqrt[2 Pi] (Cosh[c^2/(4 f) - g] FresnelC[(c + 2 f Sqrt[z])/ (Sqrt[I f] Sqrt[2 Pi])] - I FresnelS[(c + 2 f Sqrt[z])/ (Sqrt[I f] Sqrt[2 Pi])] Sinh[c^2/(4 f) - g]) + 2 I Sqrt[I f] Sinh[g + c Sqrt[z] + f z])) + 2^(-1 - m) Sum[Binomial[m, k] ((1/((-I) f)^(3/2)) ((I c - 2 b k + b m) Sqrt[2 Pi] Cos[I g + (I ((-I) c + 2 b k - b m)^2)/(4 f)] FresnelC[((-I) c + 2 b k - b m - 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] - ((-I) c + 2 b k - b m) Sqrt[2 Pi] FresnelS[((-I) c + 2 b k - b m - 2 I f Sqrt[z])/(Sqrt[(-I) f] Sqrt[2 Pi])] Sin[I g + (I ((-I) c + 2 b k - b m)^2)/(4 f)] - 2 I Sqrt[(-I) f] Sinh[g + I ((-I) c + 2 b k - b m) Sqrt[z] + f z]) + (1/(I f)^(3/2)) (((-I) c - 2 b k + b m) Sqrt[2 Pi] Cosh[g + (I c + 2 b k - b m)^2/(4 f)] FresnelC[ (I c + 2 b k - b m + 2 I f Sqrt[z])/(Sqrt[I f] Sqrt[2 Pi])] + I (I c + 2 b k - b m) Sqrt[2 Pi] FresnelS[ (I c + 2 b k - b m + 2 I f Sqrt[z])/(Sqrt[I f] Sqrt[2 Pi])] Sinh[g + (I c + 2 b k - b m)^2/(4 f)] + 2 I Sqrt[I f] Sinh[g - I (I c + 2 b k - b m) Sqrt[z] + f z])), {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0

 Standard Form

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" ", "m"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]]]], ")"]], "+", FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "f"]], ")"]], RowBox[List["3", "/", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "-", RowBox[List["2", " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["Cosh", "[", RowBox[List["g", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]], " ", RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "f", " ", SqrtBox["z"]]]]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]], " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], "2"], RowBox[List["4", " ", "f"]]]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "f"]]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]], ")"]], " ", SqrtBox["z"]]], "+", RowBox[List["f", " ", "z"]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cos m ( b z ) cosh ( z c + g + f z ) z 1 ( f ) 3 / 2 ( 2 - m - 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( c 2 π ( cosh ( c 2 4 f - g ) C TagBox["C", FresnelC] ( c + 2 f z f 2 π ) - S TagBox["S", FresnelS] ( c + 2 f z f 2 π ) sinh ( c 2 4 f - g ) ) + 2 f sinh ( z c + g + f z ) ) ) + 2 - m - 1 k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 ( - f ) 3 / 2 ( ( c - 2 b k + b m ) 2 π cos ( ( - c + 2 b k - b m ) 2 4 f + g ) C TagBox["C", FresnelC] ( - c + 2 b k - b m - 2 f z - f 2 π ) - ( - c + 2 b k - b m ) 2 π S TagBox["S", FresnelS] ( - c + 2 b k - b m - 2 f z - f 2 π ) sin ( ( - c + 2 b k - b m ) 2 4 f + g ) - 2 - f sinh ( g + f z + ( - c + 2 b k - b m ) z ) ) + 1 ( f ) 3 / 2 ( ( - c - 2 b k + b m ) 2 π cosh ( ( c + 2 b k - b m ) 2 4 f + g ) C TagBox["C", FresnelC] ( c + 2 b k - b m + 2 f z f 2 π ) + ( c + 2 b k - b m ) 2 π S TagBox["S", FresnelS] ( c + 2 b k - b m + 2 f z f 2 π ) sinh ( ( c + 2 b k - b m ) 2 4 f + g ) + 2 f sinh ( g + f z - ( c + 2 b k - b m ) z ) ) ) /; m + Condition z b z 1 2 m z 1 2 c g f z 1 f 3 2 -1 2 -1 m -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 c 2 1 2 c 2 4 f -1 -1 g FresnelC c 2 f z 1 2 f 1 2 2 1 2 -1 -1 FresnelS c 2 f z 1 2 f 1 2 2 1 2 -1 c 2 4 f -1 -1 g 2 f 1 2 z 1 2 c g f z 2 -1 m -1 k 0 m -1 2 -1 Binomial m k 1 -1 f 3 2 -1 c -1 2 b k b m 2 1 2 -1 c 2 b k -1 b m 2 4 f -1 g FresnelC -1 c 2 b k -1 b m -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 -1 -1 c 2 b k -1 b m 2 1 2 FresnelS -1 c 2 b k -1 b m -1 2 f z 1 2 -1 f 1 2 2 1 2 -1 -1 c 2 b k -1 b m 2 4 f -1 g -1 2 -1 f 1 2 g f z -1 c 2 b k -1 b m z 1 2 1 f 3 2 -1 -1 c -1 2 b k b m 2 1 2 c 2 b k -1 b m 2 4 f -1 g FresnelC c 2 b k -1 b m 2 f z 1 2 f 1 2 2 1 2 -1 c 2 b k -1 b m 2 1 2 FresnelS c 2 b k -1 b m 2 f z 1 2 f 1 2 2 1 2 -1 c 2 b k -1 b m 2 4 f -1 g 2 f 1 2 g f z -1 c 2 b k -1 b m z 1 2 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18