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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[a,z] > Representations through more general functions > Through Meijer G > Classical cases for the direct function itself





http://functions.wolfram.com/06.06.26.0006.01









  


  








Input Form





Gamma[a, Sqrt[z]] - Gamma[a, -Sqrt[z]] == (-2^(a - 2)) Sqrt[Pi] Sqrt[-z^2] MeijerG[{{}, {0}}, {{(a - 1)/2, a/2 - 1}, {-1}}, -(z/4)]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", SqrtBox["z"]]], "]"]], "-", RowBox[List["Gamma", "[", RowBox[List["a", ",", RowBox[List["-", SqrtBox["z"]]]]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", " ", SuperscriptBox["2", RowBox[List["a", "-", "2"]]]]], " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["a", "-", "1"]], "2"], ",", RowBox[List[FractionBox["a", "2"], "-", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["-", "1"]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", "4"]]]]], "]"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </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> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;3&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;z&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;a&quot;, &quot;2&quot;], &quot;-&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#915; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mrow> <mo> - </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </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> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mn> 0 </mn> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;3&quot;]], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;z&quot;, &quot;4&quot;]]], MeijerG, Rule[Editable, True]], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]]], List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;1&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;a&quot;, &quot;2&quot;], &quot;-&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a_", ",", SqrtBox["z_"]]], "]"]], "-", RowBox[List["Gamma", "[", RowBox[List["a_", ",", RowBox[List["-", SqrtBox["z_"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["a", "-", "2"]]]]], " ", SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "0", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["a", "-", "1"]], "2"], ",", RowBox[List[FractionBox["a", "2"], "-", "1"]]]], "}"]], ",", RowBox[List["{", RowBox[List["-", "1"]], "}"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", "4"]]]]], "]"]]]]]]]]


Date Added to functions.wolfram.com (modification date)





2001-10-29