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; }

 Erf

 http://functions.wolfram.com/06.25.21.0039.01

 Input Form

 Integrate[z^3 Sin[b z^2] Erf[a z], z] == (1/(4 b^2)) ((1/(2 Sqrt[Pi])) (a b z^3 ((-Sqrt[Pi] - (2 Sqrt[(a^2 - I b) z^2])/E^((a^2 - I b) z^2) + Sqrt[Pi] Erf[Sqrt[(a^2 - I b) z^2]])/((a^2 - I b) z^2)^(3/2) + (-Sqrt[Pi] - (2 Sqrt[(a^2 + I b) z^2])/E^((a^2 + I b) z^2) + Sqrt[Pi] Erf[Sqrt[(a^2 + I b) z^2]])/((a^2 + I b) z^2)^(3/2))) + (1/(a^4 + b^2)) (a (((-I) a^2 - b) Sqrt[a^2 + I b] Erf[Sqrt[a^2 + I b] z] + Sqrt[a^2 - I b] (a^2 + I b) Erfi[((I a^2 + b) z)/Sqrt[a^2 - I b]])) - 2 Erf[a z] (b z^2 Cos[b z^2] - Sin[b z^2]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox["b", "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], RowBox[List["(", RowBox[List["a", " ", "b", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SqrtBox["\[Pi]"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SqrtBox["\[Pi]"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], ")"]], "/", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]]], ")"]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "2"]]]], RowBox[List["(", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "-", "b"]], ")"]], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", RowBox[List["Erf", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", "z"]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]]]], ")"]]]], ")"]]]], "-", RowBox[List["2", " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]]]], ")"]]]]]]]]

 MathML Form

 z 3 sin ( b z 2 ) erf ( a z ) z 1 4 b 2 ( 1 2 π ( a b z 3 ( ( π erf ( ( a 2 + b ) z 2 ) - π - 2 - ( a 2 + b ) z 2 ( a 2 + b ) z 2 ) / ( ( a 2 + b ) z 2 ) 3 / 2 + ( π erf ( ( a 2 - b ) z 2 ) - π - 2 - ( a 2 - b ) z 2 ( a 2 - b ) z 2 ) / ( ( a 2 - b ) z 2 ) 3 / 2 ) ) + 1 a 4 + b 2 ( a ( ( - a 2 - b ) a 2 + b erf ( a 2 + b z ) + ( a 2 + b ) a 2 - b erfi ( ( a 2 + b ) z a 2 - b ) ) ) - 2 erf ( a z ) ( b z 2 cos ( b z 2 ) - sin ( b z 2 ) ) ) z z 3 b z 2 Erf a z 1 4 b 2 -1 1 2 1 2 -1 a b z 3 1 2 Erf a 2 b z 2 1 2 -1 1 2 -1 2 -1 a 2 b z 2 a 2 b z 2 1 2 a 2 b z 2 3 2 -1 1 2 Erf a 2 -1 b z 2 1 2 -1 1 2 -1 2 -1 a 2 -1 b z 2 a 2 -1 b z 2 1 2 a 2 -1 b z 2 3 2 -1 1 a 4 b 2 -1 a -1 a 2 -1 b a 2 b 1 2 Erf a 2 b 1 2 z a 2 b a 2 -1 b 1 2 Erfi a 2 b z a 2 -1 b 1 2 -1 -1 2 Erf a z b z 2 b z 2 -1 b z 2 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "3"], " ", RowBox[List["Sin", "[", RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "]"]], " ", RowBox[List["Erf", "[", RowBox[List["a_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List["a", " ", "b", " ", SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", SqrtBox["\[Pi]"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["-", SqrtBox["\[Pi]"]]], "-", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]]]], " ", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["Erf", "[", SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]]], "]"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["3", "/", "2"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox["\[Pi]"]]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SuperscriptBox["a", "2"]]], "-", "b"]], ")"]], " ", SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", RowBox[List["Erf", "[", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", "z"]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["a", "2"]]], "+", "b"]], ")"]], " ", "z"]], SqrtBox[RowBox[List[SuperscriptBox["a", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]]]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["a", "4"], "+", SuperscriptBox["b", "2"]]]], "-", RowBox[List["2", " ", RowBox[List["Erf", "[", RowBox[List["a", " ", "z"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cos", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], "-", RowBox[List["Sin", "[", RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "]"]]]], ")"]]]]]], RowBox[List["4", " ", SuperscriptBox["b", "2"]]]]]]]]

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

 2001-10-29