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; }

 Cos

 http://functions.wolfram.com/01.07.21.2588.01

 Input Form

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

 MathML Form

 p z 2 sin ( b z ) cos v ( c z ) z 1 p 2 - v - 2 π ( b 2 4 p ( 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]]]]] ( erf ( b + 2 p z 2 p ) + erf ( b - 2 p z 2 p ) ) - b 2 4 p ( 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]]]]] ( v mod 2 \$CellContext`v 2 ) ( erf ( b + 2 p z 2 p ) + erf ( b - 2 p z 2 p ) ) + s = 0 v - 1 2 b 2 + c 2 ( v - 2 s ) 2 2 p ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( b + 2 c s - c v ) 2 4 p erf ( b - 2 c s + c v + 2 p z 2 p ) + - ( b - 2 c s + c v ) 2 4 p erf ( b + 2 c s - c v + 2 p z 2 p ) + - ( b + 2 c s - c v ) 2 4 p erf ( b - 2 c s + c v - 2 p z 2 p ) + - ( b - 2 c s + c v ) 2 4 p erf ( b + 2 c s - c v - 2 p z 2 p ) ) ) /; v + Condition z p z 2 b z c z v 1 p 1 2 -1 2 -1 v -2 1 2 b 2 4 p -1 Binomial v v 2 -1 Erf b 2 p z 2 p 1 2 -1 Erf b -1 2 p z 2 p 1 2 -1 -1 b 2 4 p -1 Binomial v v 2 -1 \$CellContext`v 2 Erf b 2 p z 2 p 1 2 -1 Erf b -1 2 p z 2 p 1 2 -1 s 0 v -1 2 -1 b 2 c 2 v -1 2 s 2 2 p -1 Binomial v s -1 b 2 c s -1 c v 2 4 p -1 Erf b -1 2 c s c v 2 p z 2 p 1 2 -1 -1 b -1 2 c s c v 2 4 p -1 Erf b 2 c s -1 c v 2 p z 2 p 1 2 -1 -1 b 2 c s -1 c v 2 4 p -1 Erf b -1 2 c s c v -1 2 p z 2 p 1 2 -1 -1 b -1 2 c s c v 2 4 p -1 Erf b 2 c s -1 c v -1 2 p z 2 p 1 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_", " ", SuperscriptBox["z_", "2"]]]], " ", RowBox[List["Sin", "[", RowBox[List["b_", " ", "z_"]], "]"]], " ", SuperscriptBox[RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]], "v_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "v"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "p"]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]], "+", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "p"]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]], "+", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], ")"]], " ", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], "+", 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[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List[SuperscriptBox["b", "2"], "+", RowBox[List[SuperscriptBox["c", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]], "2"]]]]], RowBox[List["2", " ", "p"]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]]]], " ", RowBox[List["Erf", "[", FractionBox[RowBox[List["b", "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], RowBox[List["2", " ", SqrtBox["p"]]]], "]"]]]]]], ")"]]]]]]]], ")"]]]], SqrtBox["p"]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18