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variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z,a,b] > Series representations > Generalized power series > Expansions at generic point a==a0 > For the function itself





http://functions.wolfram.com/06.19.06.0066.01









  


  








Input Form





Beta[z, a, b] \[Proportional] Beta[z, Subscript[a, 0], b] + (1/Subscript[a, 0]^2) ((-z^Subscript[a, 0]) HypergeometricPFQ[ {1 - b, Subscript[a, 0], Subscript[a, 0]}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, z] + Beta[z, Subscript[a, 0], b] Log[z] Subscript[a, 0]^2) (a - Subscript[a, 0]) + (1/(2 Subscript[a, 0]^3)) (Beta[z, Subscript[a, 0], b] Log[z]^2 Subscript[a, 0]^3 + 2 z^Subscript[a, 0] (HypergeometricPFQ[{1 - b, Subscript[a, 0], Subscript[a, 0], Subscript[a, 0]}, {1 + Subscript[a, 0], 1 + Subscript[a, 0], 1 + Subscript[a, 0]}, z] - Subscript[a, 0] HypergeometricPFQ[ {1 - b, Subscript[a, 0], Subscript[a, 0]}, {1 + Subscript[a, 0], 1 + Subscript[a, 0]}, z] Log[z])) (a - Subscript[a, 0])^2 + O[(a - Subscript[a, 0])^3]








Standard Form





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MathML Form







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</apply> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <ci> Beta </ci> <ci> z </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["z", SubscriptBox["aa", "0"]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b"]], ",", SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]], " ", SubsuperscriptBox["aa", "0", "2"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]]]], SubsuperscriptBox["aa", "0", "2"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", SubscriptBox["aa", "0"], ",", "b"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"], " ", SubsuperscriptBox["aa", "0", "3"]]], "+", RowBox[List["2", " ", SuperscriptBox["z", SubscriptBox["aa", "0"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b"]], ",", SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "z"]], "]"]], "-", RowBox[List[SubscriptBox["aa", "0"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b"]], ",", SubscriptBox["aa", "0"], ",", SubscriptBox["aa", "0"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["aa", "0"]]], ",", RowBox[List["1", "+", SubscriptBox["aa", "0"]]]]], "}"]], ",", "z"]], "]"]], " ", RowBox[List["Log", "[", "z", "]"]]]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", SubsuperscriptBox["aa", "0", "3"]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], "]"]], "3"]]]]]]]


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





2007-05-02