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variants of this functions
InverseGammaRegularized






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > InverseGammaRegularized[a,z1,z2] > Differentiation > Low-order differentiation > With respect to a





http://functions.wolfram.com/06.13.20.0001.01









  


  










Input Form





D[InverseGammaRegularized[a, Subscript[z, 1], Subscript[z, 2]], a] == E^w w^(1 - a) ((1/a^2) (w^a HypergeometricPFQ[{a, a}, {1 + a, 1 + a}, -w] - Subscript[z, 1]^a HypergeometricPFQ[{a, a}, {1 + a, 1 + a}, -Subscript[z, 1]]) + Gamma[a, w, 0] Log[w] + Gamma[a, 0, Subscript[z, 1]] Log[Subscript[z, 1]] + Gamma[a, Subscript[z, 1], w] PolyGamma[a]) /; w == InverseGammaRegularized[a, Subscript[z, 1], Subscript[z, 2]]










Standard Form





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MathML Form







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</mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> w </mi> <mo> , </mo> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> w </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> w </mi> <mo> &#10869; </mo> <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> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> a </ci> </bvar> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <apply> <power /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> w </ci> <ci> a </ci> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <ci> a </ci> <ci> a </ci> </list> <list> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <ci> a </ci> <ci> a </ci> </list> <list> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> <ci> w </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ln /> <ci> w </ci> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> <cn type='integer'> 0 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ln /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> w </ci> </apply> <apply> <ci> PolyGamma </ci> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["a_"]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], " ", SuperscriptBox["w", RowBox[List["1", "-", "a"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox["w", "a"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", RowBox[List["-", "w"]]]], "]"]]]], "-", RowBox[List[SubsuperscriptBox["zz", "1", "a"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", RowBox[List["-", SubscriptBox["zz", "1"]]]]], "]"]]]]]], SuperscriptBox["a", "2"]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", "w", ",", "0"]], "]"]], " ", RowBox[List["Log", "[", "w", "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", "0", ",", SubscriptBox["zz", "1"]]], "]"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]], "+", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", SubscriptBox["zz", "1"], ",", "w"]], "]"]], " ", RowBox[List["PolyGamma", "[", "a", "]"]]]]]], ")"]]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["zz", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]]]]]]]]










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





2001-10-29





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