|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/06.28.10.0007.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Erfi[z] == -I + ((2 I)/Sqrt[Pi]) E^z^2
(1/(2 I z + 2/(2 I z + 4/(2 I z +
6/(2 I z + 8/(2 I z + 10/(2 I z + 12/(2 I z +
\[Ellipsis])))))))) /; Im[z] < 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Erfi", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", RowBox[List[FractionBox[RowBox[List["2", "\[ImaginaryI]"]], SqrtBox["\[Pi]"]], SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]], FractionBox["1", RowBox[List[RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["2", RowBox[List[RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["4", RowBox[List[RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["6", RowBox[List[RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["8", RowBox[List[RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["10", RowBox[List[RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["12", RowBox[List[RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]]]]]]], "/;", " ", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "<", "0"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> erfi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <msqrt> <mi> π </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msup> <mo> ⁢ </mo> <mfrac> <mn> 1 </mn> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mstyle scriptlevel='0'> <mfrac> <mn> 2 </mn> <mstyle scriptlevel='0'> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 4 </mn> <mstyle scriptlevel='0'> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 6 </mn> <mstyle scriptlevel='0'> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 8 </mn> <mstyle scriptlevel='0'> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 10 </mn> <mstyle scriptlevel='0'> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mn> 12 </mn> <mstyle scriptlevel='0'> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mo> … </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> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Erfi </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 10 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> <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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erfi", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", SuperscriptBox["z", "2"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["2", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["4", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["6", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["8", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["10", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "+", FractionBox["12", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "<", "0"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|