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/01.20.21.4083.01

 Input Form

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

 Standard Form

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

 MathML Form

 sinh ( z b + e ) cosh v ( z c + g ) z 2 1 - v ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) ( sinh ( z b + e ) - b z cosh ( z b + e ) ) b 2 - 2 1 - v s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 ( - b - 2 c s + c v ) 2 ( ( - b - 2 c s + c v ) z cosh ( e + 2 g s - g v - ( - b - 2 c s + c v ) z ) + sinh ( e + 2 g s - g v - ( - b - 2 c s + c v ) z ) ) + 1 ( - b + 2 c s - c v ) 2 ( ( - b + 2 c s - c v ) z cosh ( e - 2 g s + g v - ( - b + 2 c s - c v ) z ) + sinh ( e - 2 g s + g v - ( - b + 2 c s - c v ) z ) ) ) /; v + Condition z z 1 2 b e z 1 2 c g v 2 1 -1 v Binomial v v 2 -1 1 -1 \$CellContext`v 2 z 1 2 b e -1 b z 1 2 z 1 2 b e b 2 -1 -1 2 1 -1 v s 0 v -1 2 -1 Binomial v s 1 -1 b -1 2 c s c v 2 -1 -1 b -1 2 c s c v z 1 2 e 2 g s -1 g v -1 -1 b -1 2 c s c v z 1 2 e 2 g s -1 g v -1 -1 b -1 2 c s c v z 1 2 1 -1 b 2 c s -1 c v 2 -1 -1 b 2 c s -1 c v z 1 2 e -1 2 g s g v -1 -1 b 2 c s -1 c v z 1 2 e -1 2 g s g v -1 -1 b 2 c s -1 c v z 1 2 v SuperPlus [/itex]

 Rule Form

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

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

 2002-12-18