Complementary error function: Continued fraction representations (formula 06.27.10.0011)

Erfc

$Mathematica Notation$

Gamma, Beta, Erf

Continued fraction representations

Involving the function

http://functions.wolfram.com/06.27.10.0011.01

Input Form

Sqrt[(E Pi)/2] Erfc[1/Sqrt[2]] == 1/(1 + 1/(1 + 2/(1 + 3/(1 + 4/(1 + 5/(1 + \[Ellipsis]))))))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[SqrtBox[FractionBox[RowBox[List["\[ExponentialE]", " ", "\[Pi]"]], "2"]], " ", RowBox[List["Erfc", "[", FractionBox["1", SqrtBox["2"]], "]"]]]], "\[Equal]", FractionBox["1", RowBox[List["1", "+", FractionBox["1", RowBox[List["1", "+", FractionBox["2", RowBox[List["1", "+", FractionBox["3", RowBox[List["1", "+", FractionBox["4", RowBox[List["1", "+", FractionBox["5", RowBox[List["1", "+", "\[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> <msqrt> <mfrac> <mrow> <mi> ⅇ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <mrow> <mi> erfc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 3 </mn> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 4 </mn> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 5 </mn> <mrow> <mn> 1 </mn> <mo> + </mo> <mo> … </mo> </mrow> </mfrac> </mrow> </mfrac> </mrow> </mfrac> </mrow> </mfrac> </mrow> </mfrac> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <power /> <apply> <times /> <exponentiale /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> Erfc </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </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> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <ci> … </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> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SqrtBox[FractionBox[RowBox[List["\[ExponentialE]", " ", "\[Pi]"]], "2"]], " ", RowBox[List["Erfc", "[", FractionBox["1", SqrtBox["2"]], "]"]]]], "]"]], "\[RuleDelayed]", FractionBox["1", RowBox[List["1", "+", FractionBox["1", RowBox[List["1", "+", FractionBox["2", RowBox[List["1", "+", FractionBox["3", RowBox[List["1", "+", FractionBox["4", RowBox[List["1", "+", FractionBox["5", RowBox[List["1", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]]

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

2001-10-29