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/01.20.21.0814.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]], "m"], RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SqrtBox["z"]]], "+", "g"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", SuperscriptBox["c", "2"]], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]], "-", RowBox[List["c", " ", SqrtBox["z"], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["g", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], "2"]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]]]]]]]]

 MathML Form

 cos m ( b z ) cosh ( z c + g ) z 2 1 - m 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 ( c + b ( 2 k - m ) ) 2 ( cos ( g + ( c + b ( 2 k - m ) ) z ) + ( c + b ( 2 k - m ) ) z sin ( g + ( c + b ( 2 k - m ) ) z ) ) + 1 ( - c + b ( 2 k - m ) ) 2 ( cosh ( g + ( - c + b ( 2 k - m ) ) z ) - ( - c + b ( 2 k - m ) ) z sinh ( g + ( - c + b ( 2 k - m ) ) z ) ) ) - 2 1 - m ( 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 ) ( cosh ( z c + g ) - c z sinh ( z c + g ) ) c 2 /; m + Condition z b z 1 2 m z 1 2 c g 2 1 -1 m k 0 m -1 2 -1 Binomial m k 1 c b 2 k -1 m 2 -1 g c b 2 k -1 m z 1 2 c b 2 k -1 m z 1 2 g c b 2 k -1 m z 1 2 1 -1 c b 2 k -1 m 2 -1 g -1 c b 2 k -1 m z 1 2 -1 -1 c b 2 k -1 m z 1 2 g -1 c b 2 k -1 m z 1 2 -1 2 1 -1 m Binomial m m 2 -1 1 -1 \$CellContext`m 2 z 1 2 c g -1 c z 1 2 z 1 2 c g c 2 -1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SqrtBox["z_"]]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cosh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]], "-", RowBox[List["c", " ", SqrtBox["z"], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["c", " ", SqrtBox["z"]]]]], "]"]]]]]], ")"]]]], SuperscriptBox["c", "2"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Sin", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "g"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], "2"]], "+", FractionBox[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["g", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "c"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]]]], ")"]], "2"]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18