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LogGamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > LogGamma[z] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/06.11.17.0005.01









  


  








Input Form





LogGamma[-z] == Log[Pi] - Log[Sin[Pi z]] - Log[-z] - LogGamma[z] /; Inequality[-(1/2), LessEqual, Re[z], Less, Pi/2]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LogGamma", "[", RowBox[List["-", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", "\[Pi]", "]"]], "-", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["LogGamma", "[", "z", "]"]]]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "\[LessEqual]", RowBox[List["Re", "[", "z", "]"]], "<", FractionBox["\[Pi]", "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> <mstyle scriptlevel='0'> <mrow> <mi> log&#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mstyle> <mo> &#10869; </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#960; </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mstyle scriptlevel='0'> <mrow> <mi> log&#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mstyle> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8804; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LogGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <ln /> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <sin /> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> log&#915; </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <leq /> <apply> <real /> <ci> z </ci> </apply> <lt /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogGamma", "[", RowBox[List["-", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Log", "[", "\[Pi]", "]"]], "-", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["LogGamma", "[", "z", "]"]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "\[LessEqual]", RowBox[List["Re", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]]]]]]]]


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





2001-10-29