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http://functions.wolfram.com/06.11.16.0009.01
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LogGamma[1 - z] == Log[Pi] + Sign[Im[z]] Floor[Re[z]] Pi I -
Log[Sin[Pi (z - Floor[Re[z]])]] - Pi I (1 - Abs[Sign[Im[z]]])
Abs[Floor[Re[z]]] - LogGamma[z]
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Cell[BoxData[RowBox[List[RowBox[List["LogGamma", "[", RowBox[List["1", "-", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Log", "[", "\[Pi]", "]"]], "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "\[Pi]", " ", "\[ImaginaryI]"]], "-", 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", "]"]], "]"]], "]"]]]], "-", RowBox[List["LogGamma", "[", "z", "]"]]]]]]]]
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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> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mo> ⌊ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> π </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⌋ </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sgn </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Im </mi> <mo> ⁡ </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> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </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> <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>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogGamma", "[", RowBox[List["1", "-", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Log", "[", "\[Pi]", "]"]], "+", RowBox[List[RowBox[List["Sign", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], " ", RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]], "-", 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", "]"]], "]"]], "]"]]]], "-", RowBox[List["LogGamma", "[", "z", "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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