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/01.20.21.3813.01

 Input Form

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

 MathML Form

 p z cos ( b z ) cosh v ( c z ) z 2 - v - 2 ( 4 p z cos ( b z ) p + b b 2 4 p π ( erf ( b + 2 p z 2 p ) + erf ( b - 2 p z 2 p ) ) p 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 - 2 k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 4 p z ( cos ( ( b + 2 c k - c v ) z ) + cos ( ( b + c ( v - 2 k ) ) z ) ) p + ( b + 2 c k - c v ) 2 4 p π ( b + 2 c k - c v ) erf ( b + c ( 2 k - v ) + 2 p z 2 p ) p 3 / 2 - ( b + c ( v - 2 k ) ) 2 4 p π ( b + c ( v - 2 k ) ) erf ( - b + c ( 2 k - v ) - 2 p z 2 p ) p 3 / 2 + ( - b - 2 c k + c v ) 2 4 p π ( b + 2 c k - c v ) erf ( b + c ( 2 k - v ) - 2 p z 2 p ) p 3 / 2 - ( b + c ( v - 2 k ) ) 2 4 p π ( b + c ( v - 2 k ) ) erfi ( b - c ( v - 2 k ) + 2 p z 2 p ) p 3 / 2 ) /; v + Condition z p z b z 1 2 c z 1 2 v 2 -1 v -2 4 p z b z 1 2 p -1 b b 2 4 p -1 1 2 Erf b 2 p z 1 2 2 p 1 2 -1 Erf b -1 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 2 -1 v -2 k 0 v -1 2 -1 Binomial v k 4 p z b 2 c k -1 c v z 1 2 b c v -1 2 k z 1 2 p -1 b 2 c k -1 c v 2 4 p -1 1 2 b 2 c k -1 c v Erf b c 2 k -1 v 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 -1 b c v -1 2 k 2 4 p -1 1 2 b c v -1 2 k Erf -1 b c 2 k -1 v -1 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 -1 b -1 2 c k c v 2 4 p -1 1 2 b 2 c k -1 c v Erf b c 2 k -1 v -1 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 -1 b c v -1 2 k 2 4 p -1 1 2 b c v -1 2 k Erfi b -1 c v -1 2 k 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 v SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["p_", " ", "z_"]]], " ", RowBox[List["Cos", "[", RowBox[List["b_", " ", SqrtBox["z_"]]], "]"]], " ", SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List["c_", " ", SqrtBox["z_"]]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SqrtBox["z"]]], "]"]]]], "p"], "+", FractionBox[RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "p"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]], "+", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], ")"]]]], SuperscriptBox["p", RowBox[List["3", "/", "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["k", "=", "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", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["4", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["p", " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "v"]]]], ")"]], " ", SqrtBox["z"]]], "]"]], "+", RowBox[List["Cos", "[", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]], ")"]], " ", SqrtBox["z"]]], "]"]]]], ")"]]]], "p"], "-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List[RowBox[List["-", "b"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], SuperscriptBox["p", RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", "v"]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "v"]]]], ")"]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], SuperscriptBox["p", RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "v"]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "c", " ", "k"]], "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", "v"]]]], ")"]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "v"]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], SuperscriptBox["p", RowBox[List["3", "/", "2"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]]]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "b"]], "-", RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", "p", " ", SqrtBox["z"]]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], SuperscriptBox["p", 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