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/01.20.21.4048.01

 Input Form

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

 MathML Form

 sinh ( z b + e ) cosh v ( c z ) z 2 - v - z b - e ( z b + 2 ( z b + e ) ( b z - 1 ) + 1 ) ( 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 ) b 2 + 2 - v - 2 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]]]]] ( e ( 4 - e cosh ( z b + e + c ( 2 s - v ) z ) c ( 2 s - v ) + b b 2 4 c ( v - 2 s ) π erfi ( 2 c ( v - 2 s ) z - b 2 - c ( v - 2 s ) ) ( - c ( v - 2 s ) ) 3 / 2 - b - b 2 4 c ( v - 2 s ) - 2 e π erfi ( 2 c ( v - 2 s ) z - b 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 3 / 2 ) - - e ( 4 e cosh ( z b + e + c ( v - 2 s ) z ) c ( 2 s - v ) + b 2 e - b 2 4 c ( v - 2 s ) π erfi ( b + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 3 / 2 - b b 2 4 c ( v - 2 s ) π erfi ( b + 2 c ( v - 2 s ) z 2 - c ( v - 2 s ) ) ( - c ( v - 2 s ) ) 3 / 2 ) ) /; v + Condition z z 1 2 b e c z v 2 -1 v -1 z 1 2 b -1 e z 1 2 b 2 z 1 2 b e b z 1 2 -1 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 b 2 -1 2 -1 v -2 s 0 v -1 2 -1 Binomial v s e 4 -1 e z 1 2 b e c 2 s -1 v z c 2 s -1 v -1 b b 2 4 c v -1 2 s -1 1 2 Erfi 2 c v -1 2 s z 1 2 -1 b 2 -1 c v -1 2 s 1 2 -1 -1 c v -1 2 s 3 2 -1 -1 b -1 b 2 4 c v -1 2 s -1 -1 2 e 1 2 Erfi 2 c v -1 2 s z 1 2 -1 b 2 c v -1 2 s 1 2 -1 c v -1 2 s 3 2 -1 -1 -1 e 4 e z 1 2 b e c v -1 2 s z c 2 s -1 v -1 b 2 e -1 b 2 4 c v -1 2 s -1 1 2 Erfi b 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c v -1 2 s 3 2 -1 -1 b b 2 4 c v -1 2 s -1 1 2 Erfi b 2 c v -1 2 s z 1 2 2 -1 c v -1 2 s 1 2 -1 -1 c v -1 2 s 3 2 -1 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["c_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "v"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], "-", RowBox[List["b", " ", SqrtBox["z"]]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List["e", "+", RowBox[List["b", " ", SqrtBox["z"]]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["b", " ", SqrtBox["z"]]]]], ")"]]]], "+", RowBox[List["b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], SuperscriptBox["b", "2"]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "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[SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "e"]], "-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "e"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", "e"], " ", RowBox[List["Cosh", "[", RowBox[List["e", "+", RowBox[List["b", " ", SqrtBox["z"]]], "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", "z"]]]], "]"]]]], RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]]], "-", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["2", " ", "e"]], "-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18