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variants of this functions
Erf




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Erf[z] > Continued fraction representations > Involving the function





http://functions.wolfram.com/06.25.10.0010.01









  


  








Input Form





Erf[z] == 1 - ((2 z)/Sqrt[Pi]) (1/(E^z^2 (1 + 2 z^2 + ContinueFraction[{-2 k (2 k - 1), 4 k + 1 + 2 z^2}, {k, 1, Infinity}]))) /; Re[z] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Erf", "[", "z", "]"]], "\[Equal]", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["2", "z"]], SqrtBox["\[Pi]"]], RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], "/", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", SuperscriptBox["z", "2"]]], "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", " ", RowBox[List[RowBox[List[RowBox[List["-", "2"]], "k", RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "1"]], ")"]]]], ",", RowBox[List[RowBox[List["4", "k"]], "+", "1", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> erf </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> </mrow> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msubsup> <mrow> <msub> <mi> &#922; </mi> <mi> k </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 1 </mn> <mi> &#8734; </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Erf </ci> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <apply> <apply> <ci> Subscript </ci> <ci> &#922; </ci> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <ci> k </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <infinity /> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erf", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["ContinueFraction", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]]]], ",", RowBox[List[RowBox[List["4", " ", "k"]], "+", "1", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]], "}"]], ",", RowBox[List["{", RowBox[List["k", ",", "1", ",", "\[Infinity]"]], "}"]]]], "]"]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]


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





2001-10-29