Generalized error function: Integration (formula 06.26.21.0003)

Erf

$Mathematica Notation$

Gamma, Beta, Erf

Erf[z₁,z₂]

Integration

Indefinite integration

Involving only one direct function with respect to z₂

http://functions.wolfram.com/06.26.21.0003.01

Input Form

Integrate[Erf[Subscript[z, 1], Subscript[z, 2]], Subscript[z, 2]] == Subscript[z, 2] Erf[Subscript[z, 1], Subscript[z, 2]] + 1/(E^Subscript[z, 2]^2 Sqrt[Pi])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Erf", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["z", "2"]]]]]]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["z", "2"], RowBox[List["Erf", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "+", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SubsuperscriptBox["z", "2", "2"]]]], SqrtBox["\[Pi]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mrow> <mi> erf </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> + </mo> <mfrac> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> </msup> <msqrt> <mi> π </mi> </msqrt> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </bvar> <apply> <ci> Erf </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Erf </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["Erf", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["z_", "2"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SubscriptBox["zz", "2"], " ", RowBox[List["Erf", "[", RowBox[List[SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"]]], "]"]]]], "+", FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SubsuperscriptBox["zz", "2", "2"]]]], SqrtBox["\[Pi]"]]]]]]]]

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

2001-10-29

Erf[z]