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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/06.19.06.0067.01









  


  








Input Form





Beta[z, a, b] == Gamma[Subscript[a, 0]] z^Subscript[a, 0] Sum[Log[z]^k Sum[(1/(k - j)!) HypergeometricPFQRegularized[ {1 - b, Subscript[c, 1], Subscript[c, 2], \[Ellipsis], Subscript[c, j + 1]}, {1 + Subscript[c, 1], 1 + Subscript[c, 2], \[Ellipsis], 1 + Subscript[c, j + 1]}, z] (-(Gamma[Subscript[a, 0]]/Log[z]))^j (a - Subscript[a, 0])^k, {j, 0, k}], {k, 0, Infinity}] /; Subscript[c, 1] == Subscript[c, 2] == \[Ellipsis] == Subscript[c, k + 1] == Subscript[a, 0] && Element[k, Integers] && k >= 0








Standard Form





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MathML Form







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</mo> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <apply> <ci> InterpretationBox </ci> <ms> &#914; </ms> <ci> Beta </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <ms> z </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> , </ms> <ms> b </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 0 </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 0 </ms> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> &#8734; </ms> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <ms> log </ms> <ms> k </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> k </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> - </ms> <ms> j </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> ! </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#62387; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> OverscriptBox </ci> <ms> F </ms> <ms> ~ </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> </list> </apply> <ms> &#8289; </ms> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <ms> b </ms> </list> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 1 </ms> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 2 </ms> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> &#8230; </ms> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 1 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 2 </ms> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <ms> &#8230; </ms> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> , </ms> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> <apply> <ci> InterpretTemplate </ci> <apply> <ci> Function </ci> <list> <apply> <ci> SlotSequence </ci> <cn type='integer'> 1 </cn> </apply> </list> </apply> </apply> </apply> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> <ms> ; </ms> <apply> <ci> TagBox </ci> <ms> z </ms> <ci> HypergeometricPFQRegularized </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <true /> </apply> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> &#915; </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 0 </ms> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> j </ms> </apply> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> a </ms> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 0 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> k </ms> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 1 </ms> </apply> <ms> &#10869; </ms> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <ms> 2 </ms> </apply> <ms> &#10869; </ms> <ms> &#8230; </ms> <ms> &#10869; </ms> <apply> <ci> SubscriptBox </ci> <ms> c </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </apply> <ms> &#10869; </ms> <apply> <ci> SubscriptBox </ci> <ms> a </ms> <ms> 0 </ms> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </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[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], " ", SuperscriptBox["z", SubscriptBox["aa", "0"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b"]], ",", SubscriptBox["c", "1"], ",", SubscriptBox["c", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["c", RowBox[List["j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["c", "1"]]], ",", RowBox[List["1", "+", SubscriptBox["c", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["c", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List["Gamma", "[", SubscriptBox["aa", "0"], "]"]], RowBox[List["Log", "[", "z", "]"]]]]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "-", SubscriptBox["aa", "0"]]], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "1"], "\[Equal]", SubscriptBox["c", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["c", RowBox[List["k", "+", "1"]]], "\[Equal]", SubscriptBox["aa", "0"]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02