Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

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





http://functions.wolfram.com/06.19.21.0010.01









  


  










Input Form





Integrate[a^(\[Alpha] - 1) Beta[z, a, b], a] == (z^a a^(-1 + \[Alpha]))/(\[Alpha] - 1) - ((a^\[Alpha]/(\[Alpha] - 1)) Log[z] Gamma[\[Alpha], 0, (-a) Log[z]])/ ((-a) Log[z])^\[Alpha] + ((1 - b)/\[Alpha]) z^(a + 1) a^\[Alpha] Sum[((Pochhammer[2 - b, k] z^k)/((1 + k)^2 k!)) Hypergeometric2F1[\[Alpha], 1, 1 + \[Alpha], -(a/(1 + k))], {k, 0, Infinity}] /; Abs[z] < 1 && !(Element[-a, Integers] && -a >= 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["a", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["Beta", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", "a"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]]]], RowBox[List["\[Alpha]", "-", "1"]]], "-", RowBox[List[FractionBox[SuperscriptBox["a", "\[Alpha]"], RowBox[List["\[Alpha]", "-", "1"]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "z", "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", "z", "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "z", "]"]]]]]], "]"]]]], " ", "+", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "\[Alpha]"], SuperscriptBox["z", RowBox[List["a", "+", "1"]]], " ", SuperscriptBox["a", "\[Alpha]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], SuperscriptBox["z", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", "1", ",", RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["a", RowBox[List["1", "+", "k"]]]]]]], "]"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "a"]], "\[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> <mo> &#8747; </mo> <mrow> <mrow> <msup> <mi> a </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <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> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> a </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> a </mi> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> a </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </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> <mrow> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mn> 2 </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;2&quot;, &quot;-&quot;, &quot;b&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> k </mi> </msup> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mi> k </mi> <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> a </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;a&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> <mo> + </mo> <mfrac> <mrow> <msup> <mi> a </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> a </mi> </msup> </mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <msup> <mi> a </mi> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> , </mo> <mn> 0 </mn> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <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> a </mi> </mrow> <mo> &#8713; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> a </ci> </bvar> <apply> <times /> <apply> <power /> <ci> a </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Beta </ci> <ci> z </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <ci> a </ci> <ci> &#945; </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </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> b </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <power /> <ci> z </ci> <ci> k </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <factorial /> <ci> k </ci> </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> a </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> <times /> <apply> <power /> <ci> a </ci> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> a </ci> </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> <power /> <ci> a </ci> <ci> &#945; </ci> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ln /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <ci> &#945; </ci> <cn type='integer'> 0 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["a_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["Beta", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "a_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "a"], " ", SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]]]], RowBox[List["\[Alpha]", "-", "1"]]], "-", FractionBox[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "z", "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "z", "]"]]]]]], "]"]]]], RowBox[List["\[Alpha]", "-", "1"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", SuperscriptBox["z", RowBox[List["a", "+", "1"]]], " ", SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["2", "-", "b"]], ",", "k"]], "]"]], " ", SuperscriptBox["z", "k"]]], ")"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List["\[Alpha]", ",", "1", ",", RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["-", FractionBox["a", RowBox[List["1", "+", "k"]]]]]]], "]"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "k"]], ")"]], "2"], " ", RowBox[List["k", "!"]]]]]]]]], "\[Alpha]"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "a"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "a"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]]]










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





2001-10-29





© 1998-2014 Wolfram Research, Inc.