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variants of this functions
Gamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[a,z] > Continued fraction representations





http://functions.wolfram.com/06.06.10.0007.01









  


  








Input Form





Gamma[a, z] == Gamma[a] - z^a/(E^z (a - (a z)/(a + z + 1 - ((a + 1) z)/(a + z + 2 - ((a + 2) z)/(a + z + 3 - ((a + 3) z)/(a + z + 4 - ((a + 4) z)/(a + z + 5 + \[Ellipsis])))))))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], "-", FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]], RowBox[List["a", "-", FractionBox[RowBox[List["a", " ", "z"]], RowBox[List["a", "+", "z", "+", "1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], "z"]], RowBox[List["a", "+", "z", "+", "2", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "2"]], ")"]], " ", "z"]], RowBox[List["a", "+", "z", "+", "3", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "3"]], ")"]], "z"]], RowBox[List["a", "+", "z", "+", "4", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "4"]], ")"]], " ", "z"]], RowBox[List["a", "+", "z", "+", "5", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> </mrow> <mrow> <mi> a </mi> <mo> - </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 2 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 3 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 4 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 4 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> a </mi> <mo> + </mo> <mi> z </mi> <mo> + </mo> <mn> 5 </mn> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Gamma </ci> <ci> a </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 4 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> z </ci> <cn type='integer'> 5 </cn> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List["a_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], "-", FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]]]], RowBox[List["a", "-", FractionBox[RowBox[List["a", " ", "z"]], RowBox[List["a", "+", "z", "+", "1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], " ", "z"]], RowBox[List["a", "+", "z", "+", "2", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "2"]], ")"]], " ", "z"]], RowBox[List["a", "+", "z", "+", "3", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "3"]], ")"]], " ", "z"]], RowBox[List["a", "+", "z", "+", "4", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "+", "4"]], ")"]], " ", "z"]], RowBox[List["a", "+", "z", "+", "5", "+", "\[Ellipsis]"]]]]]]]]]]]]]]]]]]]]]]]]


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





2001-10-29