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variants of this functions
InverseGammaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > InverseGammaRegularized[a,z] > Series representations > Generalized power series > Expansions at z==1





http://functions.wolfram.com/06.12.06.0001.01









  


  










Input Form





InverseGammaRegularized[a, z] \[Proportional] ((-(z - 1)) Gamma[a + 1])^(1/a) + (1/(a + 1)) (((-(z - 1)) Gamma[a + 1])^(1/a))^2 + ((3 a + 5)/(2 (a + 1)^2 (a + 2))) (((-(z - 1)) Gamma[a + 1])^(1/a))^3 + O[(z - 1)^(4/a)]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]]], ")"]], RowBox[List["1", "/", "a"]]], "+", RowBox[List[FractionBox["1", RowBox[List["a", "+", "1"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]]], ")"]], RowBox[List["1", "/", "a"]]], ")"]], "2"]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["3", " ", "a"]], "+", "5", " "]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", "1"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["a", "+", "2"]], ")"]]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]]]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "1"]], "]"]]]], ")"]], RowBox[List["1", "/", "a"]]], ")"]], "3"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "1"]], ")"]], RowBox[List["4", "/", "a"]]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <semantics> <mi> Q </mi> <annotation-xml encoding='MathML-Content'> <ci> GammaRegularized </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> <mo> + </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 4 </mn> <mo> / </mo> <mi> a </mi> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> a </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2001-10-29





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