Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











variants of this functions
Beta




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Beta[z,a,b] > Series representations > Generalized power series > Expansions at z==1 > For the function itself > Logarithmic cases





http://functions.wolfram.com/06.19.06.0012.01









  


  








Input Form





Beta[z, a, b] == (((-1)^(b - 1) Gamma[a])/((-b)! Gamma[a + b])) Log[1 - z] - z^a (1 - z)^b Sum[(Pochhammer[a + b, k]/Pochhammer[b, k + 1]) (1 - z)^k, {k, 0, -b - 1}] + Pochhammer[1 - a, -b] z^a Sum[((Pochhammer[a, k] Pochhammer[1 - b, k])/(k! (k - b)!)) (PolyGamma[1 + k] - PolyGamma[a + k]) (1 - z)^k, {k, 0, Infinity}] /; Abs[z - 1] < 1 && Element[-b, Integers] && -b >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["b", "-", "1"]]], " ", RowBox[List["Gamma", "[", "a", "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["-", "b"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]]], RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], "-", RowBox[List[SuperscriptBox["z", "a"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "b"]], "-", "1"]]], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b"]], ",", "k"]], "]"]]]], RowBox[List[" ", RowBox[List["Pochhammer", "[", RowBox[List["b", ",", RowBox[List["k", "+", "1"]]]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]]]]]], "+", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["-", "b"]]]], "]"]], SuperscriptBox["z", "a"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "b"]], ")"]], "!"]]]]], RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "k"]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "-", "1"]], "]"]], "<", "1"]], "\[And]", RowBox[List["Element", "[", RowBox[List[RowBox[List["-", "b"]], ",", "Integers"]], "]"]], "\[And]", RowBox[List[RowBox[List["-", "b"]], "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> b </mi> </msup> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;b&quot;, &quot;)&quot;]], RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]]], Pochhammer] </annotation> </semantics> </mfrac> </mrow> </mrow> <mo> + </mo> <mrow> <semantics> <mrow> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;a&quot;]], &quot;)&quot;]], RowBox[List[&quot;-&quot;, &quot;b&quot;]]], SuperscriptBox[&quot;z&quot;, &quot;a&quot;]]], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mfrac> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, &quot;a&quot;, &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <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[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;b&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> b </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> Beta </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <ci> Pochhammer </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <times /> <apply> <ci> Subscript </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> a </ci> </apply> </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> <ci> a </ci> <ci> k </ci> </apply> <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> <plus /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> a </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <ci> &#8469; </ci> </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[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["b", "-", "1"]]], " ", RowBox[List["Gamma", "[", "a", "]"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", "z"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["-", "b"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["a", "+", "b"]], "]"]]]]], "-", RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "b"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "b"]], "-", "1"]]], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["a", "+", "b"]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]], RowBox[List["Pochhammer", "[", RowBox[List["b", ",", RowBox[List["k", "+", "1"]]]], "]"]]]]]]], "+", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "a"]], ",", RowBox[List["-", "b"]]]], "]"]], " ", SuperscriptBox["z", "a"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List["a", ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "b"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["a", "+", "k"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "b"]], ")"]], "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["z", "-", "1"]], "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["-", "b"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "b"]], "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29