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/01.20.21.4104.01

 Input Form

 Integrate[Sinh[b z^2] Cosh[c z^2 + f z + g]^v, z] == -((1/b) (2^(-2 - v) Sqrt[Pi] Binomial[v, v/2] ((-Sqrt[-b]) Erfi[Sqrt[-b] z] - Sqrt[b] Erfi[Sqrt[b] z]) (1 - Mod[v, 2]))) + 2^(-2 - v) Sqrt[Pi] Sum[Binomial[v, s] ((E^(g (-2 s + v) + (-2 f s + f v)^2/(-4 b + 8 c s - 4 c v)) Erfi[(2 b z - (2 s - v) (f + 2 c z))/(2 Sqrt[b + c (-2 s + v)])])/ Sqrt[b + c (-2 s + v)] + (E^(g (2 s - v) - (-2 f s + f v)^2/(-4 b + 8 c s - 4 c v)) Sqrt[-b + 2 c s - c v] Erfi[(-2 b z + (2 s - v) (f + 2 c z))/ (2 Sqrt[-b + 2 c s - c v])])/(b + c (-2 s + v)) + (1/(b + 2 c s - c v)) (E^(g (-2 s + v) - (f^2 (-2 s + v)^2)/ (-4 b - 8 c s + 4 c v)) (Sqrt[-b - 2 c s + c v] Erfi[(-2 b z - (2 s - v) (f + 2 c z))/(2 Sqrt[-b - 2 c s + c v])] + E^(2 (g (2 s - v) + (-2 f s + f v)^2/(-4 b - 8 c s + 4 c v))) Sqrt[b + 2 c s - c v] Erfi[(2 b z + (2 s - v) (f + 2 c z))/(2 Sqrt[b + 2 c s - c v])]))), {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["b", " ", SuperscriptBox["z", "2"]]], "]"]], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List[FractionBox["1", "b"], RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["-", "b"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["-", "b"]]], " ", "z"]], "]"]]]], "-", RowBox[List[SqrtBox["b"], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox["b"], " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], ")"]]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", 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[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "f", " ", "s"]], "+", RowBox[List["f", " ", "v"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "f", " ", "s"]], "+", RowBox[List["f", " ", "v"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "f", " ", "s"]], "+", RowBox[List["f", " ", "v"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh ( b z 2 ) cosh v ( c z 2 + f z + g ) z 2 - v - 2 π 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]]]]] ( g ( 2 s - v ) - ( f v - 2 f s ) 2 - 4 b + 8 c s - 4 c v - b + 2 c s - c v erfi ( ( 2 s - v ) ( f + 2 c z ) - 2 b z 2 - b + 2 c s - c v ) b + c ( v - 2 s ) + 1 b + 2 c s - c v ( g ( v - 2 s ) - f 2 ( v - 2 s ) 2 - 4 b - 8 c s + 4 c v ( - b - 2 c s + c v erfi ( - 2 b z - ( 2 s - v ) ( f + 2 c z ) 2 - b - 2 c s + c v ) + 2 ( ( f v - 2 f s ) 2 - 4 b - 8 c s + 4 c v + g ( 2 s - v ) ) b + 2 c s - c v erfi ( 2 b z + ( 2 s - v ) ( f + 2 c z ) 2 b + 2 c s - c v ) ) ) + ( f v - 2 f s ) 2 - 4 b + 8 c s - 4 c v + g ( v - 2 s ) erfi ( 2 b z - ( 2 s - v ) ( f + 2 c z ) 2 b + c ( v - 2 s ) ) b + c ( v - 2 s ) ) - 2 - v - 2 π ( 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]]]]] ( - - b erfi ( - b z ) - b erfi ( b z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) b /; v + Condition z b z 2 c z 2 f z g v 2 -1 v -2 1 2 s 0 v -1 2 -1 Binomial v s g 2 s -1 v -1 f v -1 2 f s 2 -4 b 8 c s -1 4 c v -1 -1 b 2 c s -1 c v 1 2 Erfi 2 s -1 v f 2 c z -1 2 b z 2 -1 b 2 c s -1 c v 1 2 -1 b c v -1 2 s -1 1 b 2 c s -1 c v -1 g v -1 2 s -1 f 2 v -1 2 s 2 -4 b -1 8 c s 4 c v -1 -1 b -1 2 c s c v 1 2 Erfi -2 b z -1 2 s -1 v f 2 c z 2 -1 b -1 2 c s c v 1 2 -1 2 f v -1 2 f s 2 -4 b -1 8 c s 4 c v -1 g 2 s -1 v b 2 c s -1 c v 1 2 Erfi 2 b z 2 s -1 v f 2 c z 2 b 2 c s -1 c v 1 2 -1 f v -1 2 f s 2 -4 b 8 c s -1 4 c v -1 g v -1 2 s Erfi 2 b z -1 2 s -1 v f 2 c z 2 b c v -1 2 s 1 2 -1 b c v -1 2 s 1 2 -1 -1 2 -1 v -2 1 2 Binomial v v 2 -1 -1 -1 b 1 2 Erfi -1 b 1 2 z -1 b 1 2 Erfi b 1 2 z 1 -1 \$CellContext`v 2 b -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["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SqrtBox[RowBox[List["-", "b"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox[RowBox[List["-", "b"]]], " ", "z"]], "]"]]]], "-", RowBox[List[SqrtBox["b"], " ", RowBox[List["Erfi", "[", RowBox[List[SqrtBox["b"], " ", "z"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "b"]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", 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[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "f", " ", "s"]], "+", RowBox[List["f", " ", "v"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "f", " ", "s"]], "+", RowBox[List["f", " ", "v"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "+", RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]], RowBox[List["b", "+", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", FractionBox[RowBox[List[SuperscriptBox["f", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "z"]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["g", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "f", " ", "s"]], "+", RowBox[List["f", " ", "v"]]]], ")"]], "2"], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "b"]], "-", RowBox[List["8", " ", "c", " ", "s"]], "+", RowBox[List["4", " ", "c", " ", "v"]]]]]]], ")"]]]]], " ", SqrtBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "c", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18