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http://functions.wolfram.com/06.22.20.0016.01
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D[BetaRegularized[Subscript[z, 1], Subscript[z, 2], a, b],
{Subscript[z, 2], \[Alpha]}] ==
(1/Beta[a, b]) Sum[(Pochhammer[1 - b, k] FDPowerConstant[Subscript[z, 2],
k + a, \[Alpha]] Subscript[z, 2]^(k + a - \[Alpha]))/((a + k) k!),
{k, 0, Infinity}] - BetaRegularized[Subscript[z, 1], a, b]
(1/(Subscript[z, 2]^\[Alpha] Gamma[1 - \[Alpha]])) /;
Abs[Subscript[z, 2]] < 1
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List[SubscriptBox["z", "2"], ",", "\[Alpha]"]], "}"]]], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["z", "2"], ",", RowBox[List["k", "+", "a"]], ",", "\[Alpha]"]], "]"]], SubsuperscriptBox["z", "2", RowBox[List["k", "+", "a", "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "-", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", "a", ",", "b"]], "]"]], FractionBox[SubsuperscriptBox["z", "2", RowBox[List["-", "\[Alpha]"]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "2"], "]"]], "<", "1"]]]]]]
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<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> ∂ </mo> <mi> α </mi> </msup> <mrow> <msub> <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> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> ∂ </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mi> α </mi> </msubsup> </mrow> </mfrac> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <semantics> <mi> Β </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> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], "k"], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <msubsup> <mi> ℱ𝒞 </mi> <mi> exp </mi> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> - </mo> <mi> α </mi> </mrow> </msubsup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> α </mi> </mrow> </msubsup> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> I </mi> <annotation-xml encoding='MathML-Content'> <ci> BetaRegularized </ci> </annotation-xml> </semantics> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> D </ci> <apply> <ci> BetaRegularized </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'> 2 </cn> </apply> <ci> α </ci> </list> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> ℱ𝒞 </ci> <ci> exp </ci> </apply> <ci> α </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <ci> a </ci> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BetaRegularized </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List[SubscriptBox["z_", "2"], ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "k"]], "]"]], " ", RowBox[List["FDPowerConstant", "[", RowBox[List[SubscriptBox["zz", "2"], ",", RowBox[List["k", "+", "a"]], ",", "\[Alpha]"]], "]"]], " ", SubsuperscriptBox["zz", "2", RowBox[List["k", "+", "a", "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["(", RowBox[List["a", "+", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]], RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]]], "-", FractionBox[RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "1"], ",", "a", ",", "b"]], "]"]], " ", SubsuperscriptBox["zz", "2", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "2"], "]"]], "<", "1"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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