Im[Erfi[x + I y]] == (x/(2 y)) Sqrt[-(y^2/x^2)] (Erfi[x - x Sqrt[-(y^2/x^2)]] - Erfi[x + x Sqrt[-(y^2/x^2)]])

Cell[BoxData[RowBox[List[RowBox[List["Im", "[", RowBox[List["Erfi", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[FractionBox["x", RowBox[List["2", "y"]]], SqrtBox[RowBox[List["-", FractionBox[RowBox[List[" ", SuperscriptBox["y", "2"]]], SuperscriptBox["x", "2"]]]]], RowBox[List["(", RowBox[List[RowBox[List["Erfi", "[", RowBox[List["x", "-", RowBox[List["x", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]], "-", RowBox[List["Erfi", "[", RowBox[List["x", "+", RowBox[List["x", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]]]], ")"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> x </mi> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mfrac> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <imaginary /> <apply> <ci> Erfi </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <ci> Erfi </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Erfi </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Im", "[", RowBox[List["Erfi", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Erfi", "[", RowBox[List["x", "-", RowBox[List["x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]], "-", RowBox[List["Erfi", "[", RowBox[List["x", "+", RowBox[List["x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]]]]], "]"]]]], ")"]]]], RowBox[List["2", " ", "y"]]]]]]]

2001-10-29