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variants of this functions
InverseBetaRegularized






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.23.20.0007.01









  


  










Input Form





D[InverseBetaRegularized[z, a, b], {z, 3}] == (1 - w)^(1 - 3 b) w^(1 - 3 a) (1 + 2 a^2 (-1 + w)^2 + (-6 + 4 b) w + (6 - 7 b + 2 b^2) w^2 + a (-1 + w) (3 + (-7 + 4 b) w)) Beta[a, b]^3 /; w == InverseBetaRegularized[z, a, b]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "3"]], "}"]]], RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", RowBox[List["3", " ", "b"]]]]], " ", SuperscriptBox["w", RowBox[List["1", "-", RowBox[List["3", " ", "a"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["4", " ", "b"]]]], ")"]], " ", "w"]], "+", RowBox[List[RowBox[List["(", RowBox[List["6", "-", RowBox[List["7", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SuperscriptBox["w", "2"]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["4", " ", "b"]]]], ")"]], " ", "w"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]], "3"]]]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]]]]]]]]










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> 3 </mn> </msup> <mrow> <msubsup> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> w </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> + </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> w </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 6 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> w </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> w </mi> <mo> &#10869; </mo> <mrow> <msubsup> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </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'> 3 </cn> </degree> </bvar> <apply> <ci> InverseBetaRegularized </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> b </ci> </apply> </apply> </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'> 3 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> <cn type='integer'> -7 </cn> </apply> <ci> w </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <ci> b </ci> </apply> </apply> <cn type='integer'> 6 </cn> </apply> <apply> <power /> <ci> w </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> b </ci> </apply> <cn type='integer'> -6 </cn> </apply> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <ci> InverseBetaRegularized </ci> <ci> z </ci> <ci> a </ci> <ci> b </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_", ",", "3"]], "}"]]]]], RowBox[List["InverseBetaRegularized", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", RowBox[List["3", " ", "b"]]]]], " ", SuperscriptBox["w", RowBox[List["1", "-", RowBox[List["3", " ", "a"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "6"]], "+", RowBox[List["4", " ", "b"]]]], ")"]], " ", "w"]], "+", RowBox[List[RowBox[List["(", RowBox[List["6", "-", RowBox[List["7", " ", "b"]], "+", RowBox[List["2", " ", SuperscriptBox["b", "2"]]]]], ")"]], " ", SuperscriptBox["w", "2"]]], "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "7"]], "+", RowBox[List["4", " ", "b"]]]], ")"]], " ", "w"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]], "3"]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List["InverseBetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]]]]]]]]]]










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





2007-05-02





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