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variants of this functions
InverseGammaRegularized




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.12.20.0006.01









  


  








Input Form





D[InverseGammaRegularized[a, z], {z, 4}] == E^(4 w) w^(1 - 4 a) (1 - 6 a^3 + a^2 (11 + 18 w) + w (11 + 6 w (3 + w)) - a (6 + w (29 + 18 w))) Gamma[a]^4 /; w == InverseGammaRegularized[a, z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "4"]], "}"]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "w"]]], " ", SuperscriptBox["w", RowBox[List["1", "-", RowBox[List["4", " ", "a"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["6", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["18", " ", "w"]]]], ")"]]]], "+", RowBox[List["w", " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["6", " ", "w", " ", RowBox[List["(", RowBox[List["3", "+", "w"]], ")"]]]]]], ")"]]]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List["w", " ", RowBox[List["(", RowBox[List["29", "+", RowBox[List["18", " ", "w"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "4"]]]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]]]]]]]]








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> 4 </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> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> w </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 18 </mn> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 11 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 18 </mn> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 29 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mi> w </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 11 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> w </mi> <mo> &#63449; </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> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 4 </cn> </degree> </bvar> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> w </ci> </apply> </apply> <apply> <power /> <ci> w </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -6 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 18 </cn> <ci> w </ci> </apply> <cn type='integer'> 11 </cn> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> w </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 18 </cn> <ci> w </ci> </apply> <cn type='integer'> 29 </cn> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <ci> a </ci> </apply> </apply> <apply> <times /> <ci> w </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <ci> w </ci> <apply> <plus /> <ci> w </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> 11 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <ci> InverseGammaRegularized </ci> <ci> a </ci> <ci> z </ci> </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["z_", ",", "4"]], "}"]]]]], RowBox[List["InverseGammaRegularized", "[", RowBox[List["a_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "w"]]], " ", SuperscriptBox["w", RowBox[List["1", "-", RowBox[List["4", " ", "a"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["6", " ", SuperscriptBox["a", "3"]]], "+", RowBox[List[SuperscriptBox["a", "2"], " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["18", " ", "w"]]]], ")"]]]], "+", RowBox[List["w", " ", RowBox[List["(", RowBox[List["11", "+", RowBox[List["6", " ", "w", " ", RowBox[List["(", RowBox[List["3", "+", "w"]], ")"]]]]]], ")"]]]], "-", RowBox[List["a", " ", RowBox[List["(", RowBox[List["6", "+", RowBox[List["w", " ", RowBox[List["(", RowBox[List["29", "+", RowBox[List["18", " ", "w"]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", "a", "]"]], "4"]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List["InverseGammaRegularized", "[", RowBox[List["a", ",", "z"]], "]"]]]]]]]]]]


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





2007-05-02