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[a,b] > Representations through equivalent functions > With related functions





http://functions.wolfram.com/06.18.27.0003.01









  


  








Input Form





Beta[a, b] == 2^((1/4) (3 + Cos[2 a Pi] + Cos[2 b Pi] - Cos[2 (a + b) Pi])) Pi^((1/2) (Sin[a Pi]^2 + Sin[b Pi]^2 - Sin[(a + b) Pi]^2)) (((2 a - 2)!! (2 b - 2)!!)/(2 a + 2 b - 2)!!)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "a", " ", "\[Pi]"]], "]"]], "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "b", " ", "\[Pi]"]], "]"]], "-", RowBox[List["Cos", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "\[Pi]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], "2"], "-", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "\[Pi]"]], "]"]], "2"]]], ")"]]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "2"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "2"]], ")"]], "!!"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "a"]], "+", RowBox[List["2", "b"]], "-", "2"]], ")"]], "!!"]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> &#914; </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> <mo> &#10869; </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mrow> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> !! </mo> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <pi /> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <pi /> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> a </ci> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <sin /> <apply> <times /> <ci> b </ci> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <pi /> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Factorial2 </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> b </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Beta", "[", RowBox[List["a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "a", " ", "\[Pi]"]], "]"]], "+", RowBox[List["Cos", "[", RowBox[List["2", " ", "b", " ", "\[Pi]"]], "]"]], "-", RowBox[List["Cos", "[", RowBox[List["2", " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "\[Pi]"]], "]"]]]], ")"]]]]], " ", SuperscriptBox["\[Pi]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List["b", " ", "\[Pi]"]], "]"]], "2"], "-", SuperscriptBox[RowBox[List["Sin", "[", RowBox[List[RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", "\[Pi]"]], "]"]], "2"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "-", "2"]], ")"]], "!!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b"]], "-", "2"]], ")"]], "!!"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a"]], "+", RowBox[List["2", " ", "b"]], "-", "2"]], ")"]], "!!"]]]]]]]


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





2001-10-29