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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/06.25.10.0001.01









  


  








Input Form





Erf[z] == 1 - (1/(E^z^2 Sqrt[Pi])) (1/ (z + 1/(2 (z + 1/(z + 3/(2 (z + 2/(z + 5/(2 (z + 3/(z + \[Ellipsis]))))))))))) /; Re[z] > 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Erf", "[", "z", "]"]], "\[Equal]", RowBox[List["1", "-", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], SqrtBox["\[Pi]"]], FractionBox["1", RowBox[List["z", "+", FractionBox[RowBox[List["1", "/", "2"]], RowBox[List["z", "+", FractionBox["1", RowBox[List["z", "+", FractionBox[RowBox[List["3", "/", "2"]], RowBox[List["z", "+", FractionBox["2", RowBox[List["z", "+", FractionBox[RowBox[List["5", "/", "2"]], RowBox[List["z", "+", FractionBox["3", RowBox[List["z", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]]]]]]], "/;", " ", 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> <mrow> <mfrac> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 2 </mn> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mn> 3 </mn> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </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 /> <apply> <times /> <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> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["1", RowBox[List["2", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["1", RowBox[List["z", "+", FractionBox["3", RowBox[List["2", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["2", RowBox[List["z", "+", FractionBox["5", RowBox[List["2", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["3", RowBox[List["z", "+", "\[Ellipsis]"]]]]], ")"]]]]]]]]]], ")"]]]]]]]]]], ")"]]]]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]


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





2001-10-29