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variants of this functions
Erf




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erf[z1,z2] > Integration > Indefinite integration > Involving one direct function and elementary functions with respect to z2 > Involving power function





http://functions.wolfram.com/06.26.21.0004.01









  


  








Input Form





Integrate[Subscript[z, 2]^(\[Alpha] - 1) Erf[Subscript[z, 1], Subscript[z, 2]], Subscript[z, 2]] == (Subscript[z, 2]^\[Alpha]/\[Alpha]) Erf[Subscript[z, 1], Subscript[z, 2]] + (Subscript[z, 2]^(1 + \[Alpha])/((Subscript[z, 2]^2)^((1 + \[Alpha])/2) (Sqrt[Pi] \[Alpha]))) Gamma[(1 + \[Alpha])/2, Subscript[z, 2]^2]








Standard Form





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








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> &#8747; </mo> <mrow> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8290; </mo> <mrow> <mi> erf </mi> <mo> &#8289; </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> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mi> &#945; </mi> </msubsup> <mtext> </mtext> </mrow> <mi> &#945; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mi> erf </mi> <mo> &#8289; </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> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mi> &#945; </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </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> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <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> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#945; </ci> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <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> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2001-10-29