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/01.07.21.2590.01

 Input Form

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

 MathML Form

 p z sin ( b z 2 ) cos v ( c z ) z 1 b ( 2 - v - 2 π ( - p 2 4 b ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b erfi ( p - 2 b z 2 - b ) + b p 2 2 b erfi ( p + 2 b z 2 b ) ) ( v mod 2 \$CellContext`v 2 - 1 ) - s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - b - ( p + c ( v - 2 s ) ) 2 4 b erfi ( p + c ( v - 2 s ) - 2 b z 2 - b ) + - b - ( p - c ( v - 2 s ) ) 2 4 b erfi ( p - ( - 2 c s + c v + 2 b z ) 2 - b ) + b ( p + c ( v - 2 s ) ) 2 4 b erfi ( p + ( - 2 c s + c v + 2 b z ) 2 b ) + b ( p - c ( v - 2 s ) ) 2 4 b erfi ( p + ( 2 c s - c v + 2 b z ) 2 b ) ) ) ) /; v + Condition z p z b z 2 c z v 1 b -1 2 -1 v -2 1 2 -1 p 2 4 b -1 Binomial v v 2 -1 -1 b 1 2 Erfi p -1 2 b z 2 -1 b 1 2 -1 b 1 2 p 2 2 b -1 Erfi p 2 b z 2 b 1 2 -1 \$CellContext`v 2 -1 -1 s 0 v -1 2 -1 Binomial v s -1 b 1 2 -1 p c v -1 2 s 2 4 b -1 Erfi p c v -1 2 s -1 2 b z 2 -1 b 1 2 -1 -1 b 1 2 -1 p -1 c v -1 2 s 2 4 b -1 Erfi p -1 -2 c s c v 2 b z 2 -1 b 1 2 -1 b 1 2 p c v -1 2 s 2 4 b -1 Erfi p -2 c s c v 2 b z 2 b 1 2 -1 b 1 2 p -1 c v -1 2 s 2 4 b -1 Erfi p 2 c s -1 c v 2 b z 2 b 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_", " ", "z_"]]], " ", RowBox[List["Sin", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", 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]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["p", "2"]]], RowBox[List["2", " ", "b"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", 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[RowBox[List["Binomial", "[", RowBox[List["v", ",", "s"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "b", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "c", " ", "s"]], "-", RowBox[List["c", " ", "v"]], "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]], "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "b"]]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", "c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]]]], ")"]], "2"]]], RowBox[List["4", " ", "b"]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["p", "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]], "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]]]], ")"]]]]]]]], ")"]]]], "b"], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "&&", RowBox[List["v", ">", "0"]]]]]]]]]]

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

 2002-12-18