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/01.20.21.4621.01

 Input Form

 Integrate[E^(p Sqrt[z]) Sinh[b z] Cosh[c Sqrt[z]]^v, z] == 2^(-1 - v) Binomial[v, v/2] ((2 E^(p Sqrt[z]) Cosh[b z])/b + (E^(p^2/(4 b)) p Sqrt[Pi] Erfi[(p - 2 b Sqrt[z])/(2 Sqrt[-b])])/ (2 (-b)^(3/2)) - (p Sqrt[Pi] Erfi[(p + 2 b Sqrt[z])/(2 Sqrt[b])])/ (E^(p^2/(4 b)) (2 b^(3/2)))) (1 - Mod[v, 2]) + 2^(-1 - v) Sum[Binomial[v, s] ((E^(b z) (E^((p - c (-2 s + v)) Sqrt[z]) + E^((p + c (-2 s + v)) Sqrt[z])) (1 + E^(-2 b z)))/b + (1/(2 (-b)^(3/2))) (E^((p - c (-2 s + v))^2/(4 b)) Sqrt[Pi] (p - c (-2 s + v)) Erfi[(p - c (-2 s + v) - 2 b Sqrt[z])/ (2 Sqrt[-b])]) + (1/(2 (-b)^(3/2))) (E^((p + c (-2 s + v))^2/(4 b)) Sqrt[Pi] (p + c (-2 s + v)) Erfi[(p + c (-2 s + v) - 2 b Sqrt[z])/(2 Sqrt[-b])]) - (1/(2 b^(3/2))) ((Sqrt[Pi] (p - c (-2 s + v)) Erfi[(p - c (-2 s + v) + 2 b Sqrt[z])/(2 Sqrt[b])])/ E^((p - c (-2 s + v))^2/(4 b))) - (1/(2 b^(3/2))) ((Sqrt[Pi] (p + c (-2 s + v)) Erfi[(p + c (-2 s + v) + 2 b Sqrt[z])/ (2 Sqrt[b])])/E^((p + c (-2 s + v))^2/(4 b)))), {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[SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["Sinh", "[", RowBox[List["b", " ", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c", " ", SqrtBox["z"]]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", SqrtBox["z"]]]], " ", RowBox[List["Cosh", "[", RowBox[List["b", " ", "z"]], "]"]]]], "b"], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["p", "2"], RowBox[List["4", " ", "b"]]]], " ", "p", " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", 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"2"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"], RowBox[List["4", " ", "b"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", SuperscriptBox["b", RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"], RowBox[List["4", " ", "b"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["b"]]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 p z sinh ( b z ) cosh v ( c z ) z 2 - v - 1 ( 2 p z cosh ( b z ) b + p 2 4 b p π erfi ( p - 2 b z 2 - b ) 2 ( - b ) 3 / 2 - - p 2 4 b p π erfi ( 2 z b + p 2 b ) 2 b 3 / 2 ) ( 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 ) + 2 - v - 1 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]]]]] ( b z ( ( p - c ( v - 2 s ) ) z + ( p + c ( v - 2 s ) ) z ) ( 1 + - 2 b z ) b + ( p + c ( v - 2 s ) ) 2 4 b π ( p + c ( v - 2 s ) ) erfi ( - 2 z b + p + c ( v - 2 s ) 2 - b ) 2 ( - b ) 3 / 2 + ( p - c ( v - 2 s ) ) 2 4 b π ( p - c ( v - 2 s ) ) erfi ( - 2 z b + p - c ( v - 2 s ) 2 - b ) 2 ( - b ) 3 / 2 - - ( p + c ( v - 2 s ) ) 2 4 b π ( p + c ( v - 2 s ) ) erfi ( 2 z b + p + c ( v - 2 s ) 2 b ) 2 b 3 / 2 - - ( p - c ( v - 2 s ) ) 2 4 b π ( p - c ( v - 2 s ) ) erfi ( 2 z b + p - c ( v - 2 s ) 2 b ) 2 b 3 / 2 ) /; v + Condition z p z 1 2 b z c z 1 2 v 2 -1 v -1 2 p z 1 2 b z b -1 p 2 4 b -1 p 1 2 Erfi p -1 2 b z 1 2 2 -1 b 1 2 -1 2 -1 b 3 2 -1 -1 -1 p 2 4 b -1 p 1 2 Erfi 2 z 1 2 b p 2 b 1 2 -1 2 b 3 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 2 -1 v -1 s 0 v -1 2 -1 Binomial v s b z p -1 c v -1 2 s z 1 2 p c v -1 2 s z 1 2 1 -2 b z b -1 p c v -1 2 s 2 4 b -1 1 2 p c v -1 2 s Erfi -2 z 1 2 b p c v -1 2 s 2 -1 b 1 2 -1 2 -1 b 3 2 -1 p -1 c v -1 2 s 2 4 b -1 1 2 p -1 c v -1 2 s Erfi -2 z 1 2 b p -1 c v -1 2 s 2 -1 b 1 2 -1 2 -1 b 3 2 -1 -1 -1 p c v -1 2 s 2 4 b -1 1 2 p c v -1 2 s Erfi 2 z 1 2 b p c v -1 2 s 2 b 1 2 -1 2 b 3 2 -1 -1 -1 p -1 c v -1 2 s 2 4 b -1 1 2 p -1 c v -1 2 s Erfi 2 z 1 2 b p -1 c v -1 2 s 2 b 1 2 -1 2 b 3 2 -1 v SuperPlus [/itex]

 Rule Form

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

 2002-12-18