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 | | http://functions.wolfram.com/06.06.26.0014.01 | 
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 | | E^((Pi I a)/2) Gamma[a, -z] - Gamma[a, z]/E^((Pi I a)/2) == 
  I 2^a Sqrt[Pi] MeijerG[{{}, {1}}, {{0, (1 + a)/2}, {a/2}}, -((I z)/2), 
    1/2] /; Inequality[0, Less, Arg[z], LessEqual, Pi] | 
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 | | Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "a"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", ",", RowBox[List["-", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "a"]], "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", ",", "z"]], "]"]]]]]], "\[Equal]", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", "a"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox[RowBox[List["1", "+", "a"]], "2"]]], "}"]], ",", RowBox[List["{", FractionBox["a", "2"], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], ",", FractionBox["1", "2"]]], "]"]]]]]], "/;", RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]]]] | 
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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>  <mrow>  <mrow>  <msup>  <mi> ⅇ </mi>  <mfrac>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mrow>  <mo> - </mo>  <mi> z </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <msup>  <mi> ⅇ </mi>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msup>  <mn> 2 </mn>  <mi> a </mi>  </msup>  <mo> ⁢ </mo>  <msqrt>  <mi> π </mi>  </msqrt>  <mo> ⁢ </mo>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 1 </mn>  <mo> , </mo>  <mn> 3 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mn> 1 </mn>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mfrac>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> , </mo>  <mfrac>  <mi> a </mi>  <mn> 2 </mn>  </mfrac>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["2", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[TagBox["1", MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "1"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["a", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mn> 0 </mn>  <mo> < </mo>  <mrow>  <mi> arg </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ≤ </mo>  <mi> π </mi>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <mrow>  <mrow>  <mrow>  <mrow>  <msup>  <mi> ⅇ </mi>  <mfrac>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mrow>  <mo> - </mo>  <mi> z </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <msup>  <mi> ⅇ </mi>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mi> π </mi>  <mo> ⁢ </mo>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> a </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> Γ </mi>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <msup>  <mn> 2 </mn>  <mi> a </mi>  </msup>  <mo> ⁢ </mo>  <msqrt>  <mi> π </mi>  </msqrt>  <mo> ⁢ </mo>  <semantics>  <mrow>  <msubsup>  <mi> G </mi>  <mrow>  <mn> 1 </mn>  <mo> , </mo>  <mn> 3 </mn>  </mrow>  <mrow>  <mn> 2 </mn>  <mo> , </mo>  <mn> 0 </mn>  </mrow>  </msubsup>  <mo> ⁡ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mrow>  <mo> - </mo>  <mfrac>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> , </mo>  <mfrac>  <mn> 1 </mn>  <mn> 2 </mn>  </mfrac>  </mrow>  <mo> ❘ </mo>  <mtable>  <mtr>  <mtd>  <mn> 1 </mn>  </mtd>  </mtr>  <mtr>  <mtd>  <mrow>  <mn> 0 </mn>  <mo> , </mo>  <mfrac>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mn> 1 </mn>  </mrow>  <mn> 2 </mn>  </mfrac>  <mo> , </mo>  <mfrac>  <mi> a </mi>  <mn> 2 </mn>  </mfrac>  </mrow>  </mtd>  </mtr>  </mtable>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["1", ",", "3"]], RowBox[List["2", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "2"]]], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["1", "2"], MeijerG, Rule[Editable, True]]]], MeijerG], "\[VerticalSeparator]", GridBox[List[List[TagBox["1", MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox["0", MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox[RowBox[List["a", "+", "1"]], "2"], MeijerG, Rule[Editable, True]], ",", TagBox[FractionBox["a", "2"], MeijerG, Rule[Editable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False]] </annotation>  </semantics>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mn> 0 </mn>  <mo> < </mo>  <mrow>  <mi> arg </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ≤ </mo>  <mi> π </mi>  </mrow>  </mrow>  </annotation-xml>  </semantics>  </math> 
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 | | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "a_"]], "2"]], " ", RowBox[List["Gamma", "[", RowBox[List["a_", ",", RowBox[List["-", "z_"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", "a_"]], ")"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["a_", ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["2", "a"], " ", SqrtBox["\[Pi]"], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox[RowBox[List["1", "+", "a"]], "2"]]], "}"]], ",", RowBox[List["{", FractionBox["a", "2"], "}"]]]], "}"]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]]]], ",", FractionBox["1", "2"]]], "]"]]]], "/;", RowBox[List["0", "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]]]]]] | 
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 | | Date Added to functions.wolfram.com (modification date) | 
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