Imaginary error function: Representations through equivalent functions (formula 06.28.27.0004)

Erfi

$Mathematica Notation$

Gamma, Beta, Erf

Erfi[z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/06.28.27.0004.01

Input Form

Erfi[z] == (1 - I) (FresnelC[((1 + I) z)/Sqrt[Pi]] - I FresnelS[((1 + I) z)/Sqrt[Pi]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Erfi", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], SqrtBox["\[Pi]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]]]]]]]]

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> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> C </mi> <annotation encoding='Mathematica'> TagBox["C", FresnelC] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <msqrt> <mi> π </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> S </mi> <annotation encoding='Mathematica'> TagBox["S", FresnelS] </annotation> </semantics> <mo> ( </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <msqrt> <mi> π </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Erfi </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <plus /> <apply> <ci> FresnelC </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <ci> FresnelS </ci> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erfi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["FresnelC", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], SqrtBox["\[Pi]"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["FresnelS", "[", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", "z"]], SqrtBox["\[Pi]"]], "]"]]]]]], ")"]]]]]]]]

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

2001-10-29