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] > Integration > Indefinite integration > Involving one direct function and elementary functions with respect to b > Involving power function





http://functions.wolfram.com/06.18.21.0006.01









  


  










Input Form





Integrate[b^(\[Alpha] - 1) Beta[a, b], b] == b^(\[Alpha] - 1)/(\[Alpha] - 1) - (((a - 1) b^\[Alpha])/\[Alpha]) Sum[(Pochhammer[2 - a, k]/((k + 1) (k + 1)!)) Hypergeometric2F1[\[Alpha], 1, \[Alpha] + 1, -(b/(k + 1))], {k, 0, Infinity}] /; Re[a] > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["b", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Beta", "[", RowBox[List["a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", "b"]]]]]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox["b", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["\[Alpha]", "-", "1"]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], SuperscriptBox["b", "\[Alpha]"]]], "\[Alpha]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[" ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "a"]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]], RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", "1", ",", RowBox[List["\[Alpha]", "+", "1"]], ",", RowBox[List["-", FractionBox["b", RowBox[List["k", "+", "1"]]]]]]], "]"]]]]]]]]]]]], " ", "/;", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "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> <mo> &#8747; </mo> <mrow> <msup> <mi> b </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <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> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> b </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <msup> <mi> b </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> b </mi> <mi> &#945; </mi> </msup> </mrow> <mi> &#945; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;2&quot;, &quot;-&quot;, &quot;a&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#945; </mi> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mi> b </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;\[Alpha]&quot;, Hypergeometric2F1, Rule[Editable, True]], &quot;,&quot;, TagBox[&quot;1&quot;, Hypergeometric2F1, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[RowBox[List[&quot;\[Alpha]&quot;, &quot;+&quot;, &quot;1&quot;]], Hypergeometric2F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;b&quot;, RowBox[List[&quot;k&quot;, &quot;+&quot;, &quot;1&quot;]]]]], Hypergeometric2F1, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> b </ci> </bvar> <apply> <times /> <apply> <power /> <ci> b </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Beta </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> b </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <ci> b </ci> <ci> &#945; </ci> </apply> <apply> <power /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1 </ci> <ci> &#945; </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> b </ci> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> a </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["b_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Beta", "[", RowBox[List["a_", ",", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "b_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["b", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["\[Alpha]", "-", "1"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["a", "-", "1"]], ")"]], " ", SuperscriptBox["b", "\[Alpha]"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "a"]], ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", "1", ",", RowBox[List["\[Alpha]", "+", "1"]], ",", RowBox[List["-", FractionBox["b", RowBox[List["k", "+", "1"]]]]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "!"]]]]]]]]], "\[Alpha]"]]], "/;", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", "0"]]]]]]]]










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





2001-10-29