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variants of this functions
BetaRegularized




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.22.20.0005.01









  


  








Input Form





D[BetaRegularized[Subscript[z, 1], Subscript[z, 2], a, b], a] == (-BetaRegularized[Subscript[z, 1], a, b]) Log[Subscript[z, 1]] + BetaRegularized[Subscript[z, 2], a, b] Log[Subscript[z, 2]] + BetaRegularized[Subscript[z, 1], Subscript[z, 2], a, b] (-PolyGamma[a] + PolyGamma[a + b]) + ((Gamma[a] Gamma[a + b])/Gamma[b]) (HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, Subscript[z, 1]] Subscript[z, 1]^a - HypergeometricPFQRegularized[{a, a, 1 - b}, {1 + a, 1 + a}, Subscript[z, 2]] Subscript[z, 2]^a)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "a"], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", "a", ",", "b"]], "]"]]]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]], "+", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]], "+", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyGamma", "[", "a", "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], RowBox[List["Gamma", "[", "b", "]"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["z", "1"]]], "]"]], " ", SubsuperscriptBox["z", "1", "a"]]], "-", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["z", "2"]]], "]"]], " ", SubsuperscriptBox["z", "2", "a"]]]]], ")"]]]]]]]]]]








MathML Form







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<ci> Gamma </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <ci> a </ci> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> a </ci> <ci> a </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </list> <list> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> </apply> <apply> <ci> HypergeometricPFQRegularized </ci> <list> <ci> a </ci> <ci> a </ci> <apply> <plus /> <cn type='integer'> 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["a_"]]], RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "1"], ",", "a", ",", "b"]], "]"]]]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]], "+", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "2"], ",", "a", ",", "b"]], "]"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]], "+", RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List[SubscriptBox["zz", "1"], ",", SubscriptBox["zz", "2"], ",", "a", ",", "b"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["PolyGamma", "[", "a", "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["zz", "1"]]], "]"]], " ", SubsuperscriptBox["zz", "1", "a"]]], "-", RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "a", ",", RowBox[List["1", "-", "b"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "a"]], ",", RowBox[List["1", "+", "a"]]]], "}"]], ",", SubscriptBox["zz", "2"]]], "]"]], " ", SubsuperscriptBox["zz", "2", "a"]]]]], ")"]]]], RowBox[List["Gamma", "[", "b", "]"]]]]]]]]]


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





2001-10-29