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variants of this functions
InverseBetaRegularized






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.24.20.0002.01









  


  










Input Form





D[InverseBetaRegularized[Subscript[z, 1], Subscript[z, 2], a, b], {Subscript[z, 1], 2}] == (1 - w)^(1 - 2 b) w^(1 - 2 a) (1 - Subscript[z, 1])^(-2 + b) Subscript[z, 1]^(-2 + a) ((-1 + a) (1 - w)^b w^a - (-2 + a + b) (1 - w)^b w^a Subscript[z, 1] + (1 + a (-1 + w) + (-2 + b) w) (1 - Subscript[z, 1])^b Subscript[z, 1]^a) /; w == InverseBetaRegularized[Subscript[z, 1], Subscript[z, 2], a, b]










Standard Form





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MathML Form







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</mo> <msup> <mi> w </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> - </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> a </mi> </msup> </mrow> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> w </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> w </mi> <mi> a </mi> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <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> <mi> b </mi> <mo> - </mo> <mn> 2 </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> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mi> a </mi> </msubsup> </mrow> </mrow> <mo> ) </mo> </mrow> </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> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <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> <ci> D </ci> <apply> <ci> InverseBetaRegularized </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> <ci> a </ci> <ci> b </ci> </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> <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'> 2 </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'> 2 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> b </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <ci> w </ci> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> -2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> w </ci> </apply> </apply> <ci> b </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> w </ci> <ci> a </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <plus /> <ci> w </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> b </ci> <cn type='integer'> -2 </cn> </apply> <ci> w </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <ci> b </ci> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> w </ci> <apply> <ci> InverseBetaRegularized </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> <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[SubscriptBox["z_", "1"], ",", "2"]], "}"]]]]], RowBox[List["InverseBetaRegularized", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], RowBox[List["1", "-", RowBox[List["2", " ", "b"]]]]], " ", SuperscriptBox["w", RowBox[List["1", "-", RowBox[List["2", " ", "a"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "1"]]], ")"]], RowBox[List[RowBox[List["-", "2"]], "+", "b"]]], " ", SubsuperscriptBox["zz", "1", RowBox[List[RowBox[List["-", "2"]], "+", "a"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", "a"]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "a", "+", "b"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "w"]], ")"]], "b"], " ", SuperscriptBox["w", "a"], " ", SubscriptBox["zz", "1"]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "w"]], ")"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "b"]], ")"]], " ", "w"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SubscriptBox["zz", "1"]]], ")"]], "b"], " ", SubsuperscriptBox["zz", "1", "a"]]]]], ")"]]]], "/;", RowBox[List["w", "\[Equal]", RowBox[List["InverseBetaRegularized", "[", RowBox[List[SubscriptBox["zz", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]]]]]]]]]]










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





2001-10-29





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