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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 z1





http://functions.wolfram.com/06.13.20.0004.01









  


  










Input Form





D[InverseGammaRegularized[a, Subscript[z, 1], Subscript[z, 2]], {Subscript[z, 1], 2}] == (1/Subscript[z, 1]) E^(w - 2 Subscript[z, 1]) (w/Subscript[z, 1])^(1 - 2 a) (E^w w - (a - 1) (E^w - E^Subscript[z, 1] (w/Subscript[z, 1])^a) - E^Subscript[z, 1] (w/Subscript[z, 1])^a Subscript[z, 1]) /; w == InverseGammaRegularized[a, Subscript[z, 1], Subscript[z, 2]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["z", "1"], ",", "2"]], "}"]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]], "\[Equal]", RowBox[List[FractionBox["1", SubscriptBox["z", "1"]], SuperscriptBox["\[ExponentialE]", RowBox[List["w", "-", RowBox[List["2", " ", SubscriptBox["z", "1"]]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["w", SubscriptBox["z", "1"]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "a"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], " ", "w"]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", SubscriptBox["z", "1"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["w", SubscriptBox["z", "1"]], ")"]], "a"]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", SubscriptBox["z", "1"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["w", SubscriptBox["z", "1"]], ")"]], "a"], " ", SubscriptBox["z", "1"]]]]], ")"]]]]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mn> 2 </mn> </msup> <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> <mo> &#8706; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> w </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> w </mi> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mi> w </mi> </msup> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mi> w </mi> </msup> <mo> - </mo> <mrow> <msup> <mi> &#8519; </mi> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> w </mi> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#8519; </mi> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msup> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> w </mi> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mi> a </mi> </msup> </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> <ci> D </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> <list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </list> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <ci> w </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <apply> <power /> <exponentiale /> <ci> w </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <times /> <ci> w </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> a </ci> </apply> </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[RowBox[List["{", RowBox[List[SubscriptBox["z_", "1"], ",", "2"]], "}"]]]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a_", ",", SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["w", "-", RowBox[List["2", " ", SubscriptBox["zz", "1"]]]]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["w", SubscriptBox["zz", "1"]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "a"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], " ", "w"]], "-", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", "w"], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", SubscriptBox["zz", "1"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["w", SubscriptBox["zz", "1"]], ")"]], "a"]]]]], ")"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", SubscriptBox["zz", "1"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["w", SubscriptBox["zz", "1"]], ")"]], "a"], " ", SubscriptBox["zz", "1"]]]]], ")"]]]], SubscriptBox["zz", "1"]], "/;", 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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