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LogGamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > LogGamma[z] > Transformations > Transformations and argument simplifications > Argument involving basic arithmetic operations





http://functions.wolfram.com/06.11.16.0012.01









  


  








Input Form





LogGamma[-z] == -LogGamma[z] - Log[-z] + Sign[Im[z]] Floor[Re[z]] Pi I + Log[Pi] - Log[Sin[Pi (z - Floor[Re[z]])]] - Pi I (1 - Abs[Sign[Im[z]]]) Abs[Floor[Re[z]]]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["LogGamma", "[", RowBox[List["-", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["LogGamma", "[", "z", "]"]]]], "-", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", "\[Pi]", "]"]], "-", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], ")"]]]], "]"]], " ", "]"]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["Sign", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "]"]]]], ")"]], RowBox[List["Abs", "[", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> log&#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#960; </mi> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mo> &#8970; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <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> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log&#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sgn </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LogGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <imaginaryi /> <apply> <abs /> <apply> <floor /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <apply> <ci> Sign </ci> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <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 /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> LogGamma </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <floor /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <ci> Sign </ci> <apply> <imaginary /> <ci> z </ci> </apply> </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["LogGamma", "[", "z", "]"]]]], "-", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]], "+", RowBox[List["Log", "[", "\[Pi]", "]"]], "-", RowBox[List["Log", "[", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], ")"]]]], "]"]], "]"]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Abs", "[", RowBox[List["Sign", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "]"]]]], ")"]], " ", RowBox[List["Abs", "[", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "]"]]]]]]]]]]


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





2007-05-02