Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Erf






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erf[z1,z2] > Representations through more general functions > Through hypergeometric functions > Involving hypergeometric U





http://functions.wolfram.com/06.26.26.0002.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Erf", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[SubscriptBox["z", "2"], SqrtBox[SubsuperscriptBox["z", "2", "2"]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SubsuperscriptBox["z", "2", "2"]]]], " "]], SqrtBox["\[Pi]"]], RowBox[List["HypergeometricU", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", SubsuperscriptBox["z", "2", "2"]]], "]"]]]]]], ")"]]]], "-", RowBox[List[FractionBox[SubscriptBox["z", "1"], SqrtBox[SubsuperscriptBox["z", "1", "2"]]], RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SubsuperscriptBox["z", "1", "2"]]]], " "]], SqrtBox["\[Pi]"]], RowBox[List["HypergeometricU", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", SubsuperscriptBox["z", "1", "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> <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> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <msqrt> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox[&quot;U&quot;, HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <msqrt> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox[&quot;U&quot;, HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <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> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <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> <ci> HypergeometricU </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </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> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> HypergeometricU </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2001-10-29