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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/06.25.10.0005.01









  


  










Input Form





Erf[z] == (((2 z)/Sqrt[Pi]) (1/(1 - 2 z^2 + (4 z^2)/(3 - 2 z^2 + (8 z^2)/(5 - 2 z^2 = (12 z^2)/(7 - 2 z^2 + (16 z^2)/(9 - 2 z^2 + (20 z^2)/ (11 - 2 z^2 + (24 z^2)/(13 - 2 z^2 + \[Ellipsis])))))))))/E^z^2










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Erf", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "z"]], " "]], SqrtBox["\[Pi]"]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], RowBox[List["1", "/", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "2"]]], RowBox[List["3", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["8", " ", SuperscriptBox["z", "2"]]], RowBox[List[RowBox[List["5", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], "=", FractionBox[RowBox[List["12", " ", SuperscriptBox["z", "2"]]], RowBox[List["7", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["16", " ", SuperscriptBox["z", "2"]]], RowBox[List["9", "-", RowBox[List["2", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["20", " ", SuperscriptBox["z", "2"]]], RowBox[List["11", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["24", " ", SuperscriptBox["z", "2"]]], RowBox[List["13", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], ")"]]]]]]]]]]










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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mtext> </mtext> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 3 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 5 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 7 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 9 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 20 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 11 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mfrac> <mrow> <mn> 24 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mstyle scriptlevel='0'> <mrow> <mn> 13 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <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> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Erf </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <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 /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <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> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <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> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 5 </cn> <apply> <times /> <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> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <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> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <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> <apply> <times /> <cn type='integer'> 20 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 11 </cn> <apply> <times /> <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> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 13 </cn> <apply> <times /> <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> <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> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erf", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", 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"]]], "+", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "2"]]], RowBox[List["3", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["8", " ", SuperscriptBox["z", "2"]]], RowBox[List[RowBox[List["5", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]], "=", FractionBox[RowBox[List["12", " ", SuperscriptBox["z", "2"]]], RowBox[List["7", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["16", " ", SuperscriptBox["z", "2"]]], RowBox[List["9", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["20", " ", SuperscriptBox["z", "2"]]], RowBox[List["11", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", FractionBox[RowBox[List["24", " ", SuperscriptBox["z", "2"]]], RowBox[List["13", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], ")"]]]]]]]]]










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





2001-10-29





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