Erf[Subscript[z, 1], Subscript[z, 2]] == Sqrt[Subscript[z, 2]^2]/Subscript[z, 2] - Sqrt[Subscript[z, 1]^2]/ Subscript[z, 1] - (Subscript[z, 2]/Sqrt[Pi]) ExpIntegralE[1/2, Subscript[z, 2]^2] + (Subscript[z, 1]/Sqrt[Pi]) ExpIntegralE[1/2, Subscript[z, 1]^2]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> erf </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, ExpIntegralE] </annotation> </semantics> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msub> <mo> ( </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mtext> </mtext> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox[&quot;E&quot;, ExpIntegralE] </annotation> </semantics> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msub> <mo> ( </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <msqrt> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </msqrt> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mfrac> <mo> - </mo> <mfrac> <msqrt> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Erf </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ExpIntegralE </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ExpIntegralE </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Erf", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[SqrtBox[SubsuperscriptBox["zz", "2", "2"]], SubscriptBox["zz", "2"]], "-", FractionBox[SqrtBox[SubsuperscriptBox["zz", "1", "2"]], SubscriptBox["zz", "1"]], "-", FractionBox[RowBox[List[SubscriptBox["zz", "2"], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[FractionBox["1", "2"], ",", SubsuperscriptBox["zz", "2", "2"]]], "]"]]]], SqrtBox["\[Pi]"]], "+", FractionBox[RowBox[List[SubscriptBox["zz", "1"], " ", RowBox[List["ExpIntegralE", "[", RowBox[List[FractionBox["1", "2"], ",", SubsuperscriptBox["zz", "1", "2"]]], "]"]]]], SqrtBox["\[Pi]"]]]]]]]]

2001-10-29