Limit[GammaRegularized[a, Subscript[x, 1] - I \[Epsilon], Subscript[z, 2]], \[Epsilon] -> Plus[0]] == GammaRegularized[a, Subscript[x, 1], Subscript[z, 2]] + (1 - E^(-2 I Pi a)) GammaRegularized[a, 0, Subscript[x, 1]] /; Subscript[x, 1] < 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", RowBox[List[SubscriptBox["x", "1"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], ",", SubscriptBox["z", "2"]]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", " ", RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["x", "1"], ",", SubscriptBox["z", "2"]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "a"]]]]], ")"]], RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", "0", ",", SubscriptBox["x", "1"]]], "]"]]]]]]]], "/;", RowBox[List[SubscriptBox["x", "1"], "<", "0"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#1013; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#1013; </mi> </mrow> </mrow> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> a </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <msub> <mi> x </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> &#1013; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> GammaRegularized </ci> <ci> a </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> &#1013; </ci> </apply> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <ci> GammaRegularized </ci> <ci> a </ci> <cn type='integer'> 0 </cn> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> GammaRegularized </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a_", ",", RowBox[List[SubscriptBox["x_", "1"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]_"]]]], ",", SubscriptBox["z_", "2"]]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["xx", "1"], ",", SubscriptBox["zz", "2"]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "a"]]]]], ")"]], " ", RowBox[List["GammaRegularized", "[", RowBox[List["a", ",", "0", ",", SubscriptBox["xx", "1"]]], "]"]]]]]], "/;", RowBox[List[SubscriptBox["xx", "1"], "<", "0"]]]]]]]]

2001-10-29