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[z1,z2,a,b] > Integration > Indefinite integration > Involving one direct function and elementary functions with respect to a > Involving power function





http://functions.wolfram.com/06.20.21.0010.01









  


  








Input Form





Integrate[a^(\[Alpha] - 1) Beta[Subscript[z, 1], Subscript[z, 2], a, b], a] == (a^(-1 + \[Alpha]) (Subscript[z, 2]^a - Subscript[z, 1]^a))/ (\[Alpha] - 1) + (a^\[Alpha]/(\[Alpha] - 1)) ((Log[Subscript[z, 1]] Gamma[\[Alpha], 0, (-a) Log[Subscript[z, 1]]])/ ((-a) Log[Subscript[z, 1]])^\[Alpha] - (Log[Subscript[z, 2]] Gamma[\[Alpha], 0, (-a) Log[Subscript[z, 2]]])/ ((-a) Log[Subscript[z, 2]])^\[Alpha]) + ((1 - b)/\[Alpha]) a^\[Alpha] Sum[((Pochhammer[2 - b, k] (Subscript[z, 2]^(k + a + 1) - Subscript[z, 1]^(k + a + 1)))/((1 + k)^2 k!)) Hypergeometric2F1[\[Alpha], 1, 1 + \[Alpha], -(a/(1 + k))], {k, 0, Infinity}] /; Abs[Subscript[z, 1]] < 1 && Abs[Subscript[z, 2]] < 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[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"], ",", "a", ",", "b"]], "]"]], RowBox[List["\[DifferentialD]", "a"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]], RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "2", "a"], "-", SubsuperscriptBox["z", "1", "a"]]], ")"]]]], RowBox[List["\[Alpha]", "-", "1"]]], "+", RowBox[List[FractionBox[SuperscriptBox["a", "\[Alpha]"], RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "1"], "]"]]]]]], "]"]]]], " ", "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["z", "2"], "]"]]]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "\[Alpha]"], 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"]], "]"]], RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["z", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]], 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", "[", SubscriptBox["z", "1"], "]"]], "<", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["z", "2"], "]"]], "<", "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> <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 encoding='Mathematica'> TagBox[&quot;\[CapitalBeta]&quot;, BetaRegularized] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> a </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mi> a </mi> </msubsup> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mi> a </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </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> <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> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> </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> <mrow> <mfrac> <mrow> <msup> <mi> a </mi> <mi> &#945; </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <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> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <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> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <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> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <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> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <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> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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'> <mrow> <mrow> <mrow> <mo> &#8747; </mo> <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 encoding='Mathematica'> TagBox[&quot;\[CapitalBeta]&quot;, BetaRegularized] </annotation> </semantics> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> a </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mi> a </mi> </msubsup> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mi> a </mi> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </mi> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> a </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> <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> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 2 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mrow> <mi> a </mi> <mo> + </mo> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msubsup> </mrow> <mo> ) </mo> </mrow> </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> <mrow> <mfrac> <mrow> <msup> <mi> a </mi> <mi> &#945; </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <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> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <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> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <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> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <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> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <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> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <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> </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[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"], ",", "a_", ",", "b_"]], "]"]]]], RowBox[List["\[DifferentialD]", "a_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["a", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "2", "a"], "-", SubsuperscriptBox["zz", "1", "a"]]], ")"]]]], RowBox[List["\[Alpha]", "-", "1"]]], "+", FractionBox[RowBox[List[SuperscriptBox["a", "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "1"], "]"]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[Alpha]", ",", "0", ",", RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", SubscriptBox["zz", "2"], "]"]]]]]], "]"]]]]]], ")"]]]], RowBox[List["\[Alpha]", "-", "1"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "b"]], ")"]], " ", 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"]], "]"]], " ", RowBox[List["(", RowBox[List[SubsuperscriptBox["zz", "2", RowBox[List["k", "+", "a", "+", "1"]]], "-", SubsuperscriptBox["zz", "1", RowBox[List["k", "+", "a", "+", "1"]]]]], ")"]]]], ")"]], " ", 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", "[", SubscriptBox["zz", "1"], "]"]], "<", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", SubscriptBox["zz", "2"], "]"]], "<", "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