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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/06.25.10.0009.01









  


  








Input Form





Erf[z] == 1 - (((2 z)/Sqrt[Pi]) (1/(1 + 2 z^2 - 2/(5 + 2 z^2 - 12/(9 + 2 z^2 - 30/(13 + 2 z^2 - 56/(17 + 2 z^2 - 90/(21 + 2 z^2 - 132/(2 5 + 2 z^2 - \[Ellipsis])))))))))/E^z^2 /; 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]"]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["z", "2"]]]], RowBox[List["1", "/", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", SuperscriptBox["z", "2"]]], "-", FractionBox["2", RowBox[List["5", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["12", RowBox[List["9", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["30", RowBox[List["13", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["56", RowBox[List["17", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["90", RowBox[List["21", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["132", RowBox[List[RowBox[List["2", "5"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], ")"]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]








MathML Form







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</mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> - </mo> </mstyle> <mstyle scriptlevel='0'> <mfrac> <mn> 30 </mn> <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> <mfrac> <mn> 56 </mn> <mstyle scriptlevel='0'> <mrow> <mn> 17 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mfrac> <mn> 90 </mn> <mstyle scriptlevel='0'> <mrow> <mn> 21 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mfrac> <mn> 132 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mstyle scriptlevel='0'> <mn> 5 </mn> </mstyle> </mrow> <mstyle scriptlevel='0'> <mo> + </mo> </mstyle> <mstyle scriptlevel='0'> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mstyle> <mstyle scriptlevel='0'> <mo> - </mo> </mstyle> <mstyle scriptlevel='0'> <mo> &#8230; </mo> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> <mo> ) </mo> </mrow> </mrow> </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 /> <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'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 5 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 30 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 13 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 17 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 90 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 21 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 132 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <cn type='integer'> 5 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#8230; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </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[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["2", RowBox[List["5", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["12", RowBox[List["9", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["30", RowBox[List["13", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["56", RowBox[List["17", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["90", RowBox[List["21", "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", FractionBox["132", RowBox[List[RowBox[List["2", " ", "5"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "-", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]


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





2001-10-29