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   http://functions.wolfram.com/06.21.06.0054.01
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    BetaRegularized[z, a, b] \[Proportional] 
  Piecewise[{{(Log[z] Sin[a Pi])/(E^(I a Pi) Pi), Arg[z] <= 0}}, 
   (E^(I a Pi) Log[z] Sin[a Pi])/Pi] /; a + b == 1 && (Abs[z] -> Infinity) 
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   Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BetaRegularized", "[", RowBox[List["z", ",", "a", ",", "b"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "\[Pi]"]]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]]]], "\[Pi]"], ",", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]]], "}"]], "}"]], ",", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Pi]"]]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]]]], "\[Pi]"]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b"]], "\[Equal]", "1"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]] 
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   <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> I </mi>  <annotation-xml encoding='MathML-Content'>  <ci> BetaRegularized </ci>  </annotation-xml>  </semantics>  <mi> z </mi>  </msub>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> , </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ∝ </mo>  <mrow>  <mo>  </mo>  <mtable>  <mtr>  <mtd>  <mfrac>  <mrow>  <msup>  <mi> ⅇ </mi>  <mrow>  <mrow>  <mo> - </mo>  <mi> ⅈ </mi>  </mrow>  <mo> ⁢ </mo>  <mi> a </mi>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mi> π </mi>  </mfrac>  </mtd>  <mtd>  <mrow>  <mrow>  <mi> arg </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ≤ </mo>  <mn> 0 </mn>  </mrow>  </mtd>  </mtr>  <mtr>  <mtd>  <mfrac>  <mrow>  <msup>  <mi> ⅇ </mi>  <mrow>  <mi> ⅈ </mi>  <mo> ⁢ </mo>  <mi> a </mi>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  </msup>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mi> z </mi>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> sin </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> ⁢ </mo>  <mi> π </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mi> π </mi>  </mfrac>  </mtd>  <mtd>  <semantics>  <mi> True </mi>  <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation>  </semantics>  </mtd>  </mtr>  </mtable>  </mrow>  </mrow>  <mo> /; </mo>  <mrow>  <mrow>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> b </mi>  </mrow>  <mo>  </mo>  <mn> 1 </mn>  </mrow>  <mo> ∧ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation>  </semantics>  <mi> z </mi>  <semantics>  <mo> ❘ </mo>  <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation>  </semantics>  </mrow>  <semantics>  <mo> → </mo>  <annotation encoding='Mathematica'> "\[Rule]" </annotation>  </semantics>  <mi> ∞ </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <ci> Condition </ci>  <apply>  <ci> Proportional </ci>  <apply>  <ci> BetaRegularized </ci>  <ci> z </ci>  <ci> a </ci>  <ci> b </ci>  </apply>  <piecewise>  <piece>  <apply>  <times />  <apply>  <power />  <exponentiale />  <apply>  <times />  <apply>  <times />  <cn type='integer'> -1 </cn>  <imaginaryi />  </apply>  <ci> a </ci>  <pi />  </apply>  </apply>  <apply>  <ln />  <ci> z </ci>  </apply>  <apply>  <sin />  <apply>  <times />  <ci> a </ci>  <pi />  </apply>  </apply>  <apply>  <power />  <pi />  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <leq />  <apply>  <arg />  <ci> z </ci>  </apply>  <cn type='integer'> 0 </cn>  </apply>  </piece>  <otherwise>  <apply>  <times />  <apply>  <power />  <exponentiale />  <apply>  <times />  <imaginaryi />  <ci> a </ci>  <pi />  </apply>  </apply>  <apply>  <ln />  <ci> z </ci>  </apply>  <apply>  <sin />  <apply>  <times />  <ci> a </ci>  <pi />  </apply>  </apply>  <apply>  <power />  <pi />  <cn type='integer'> -1 </cn>  </apply>  </apply>  </otherwise>  </piecewise>  </apply>  <apply>  <and />  <apply>  <eq />  <apply>  <plus />  <ci> a </ci>  <ci> b </ci>  </apply>  <cn type='integer'> 1 </cn>  </apply>  <apply>  <ci> Rule </ci>  <apply>  <abs />  <ci> z </ci>  </apply>  <infinity />  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BetaRegularized", "[", RowBox[List["z_", ",", "a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "a", " ", "\[Pi]"]]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]]]], "\[Pi]"], RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "0"]]], List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "a", " ", "\[Pi]"]]], " ", RowBox[List["Log", "[", "z", "]"]], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", "\[Pi]"]], "]"]]]], "\[Pi]"], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List[RowBox[List[RowBox[List["a", "+", "b"]], "\[Equal]", "1"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]  |  
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   Date Added to functions.wolfram.com (modification date)
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