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http://functions.wolfram.com/06.11.16.0011.01
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LogGamma[-z] == -LogGamma[z] + I Pi z + 2 I Pi Floor[3/4 - Arg[z]/(2 Pi)] -
Log[1 - E^(2 I Pi z)] + Log[-2 I Pi] - Log[-z] -
2 Pi I UnitStep[-Im[z]] (UnitStep[Re[z]] Floor[Re[z]] -
UnitStep[-Re[z]] Floor[-Re[z]]) - 2 Pi I Floor[-Re[z]]
KroneckerDelta[Arg[z] - Pi]
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Cell[BoxData[RowBox[List[RowBox[List["LogGamma", "[", RowBox[List["-", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["LogGamma", "[", "z", "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "z"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["3", "4"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "-", RowBox[List["Log", "[", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "z"]]]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]"]], "]"]], "-", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "-", RowBox[List["2", "Pi", " ", "I", " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Im", "[", "z", "]"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], RowBox[List["Floor", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]], "-", RowBox[List[RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List["2", "Pi", " ", "I", " ", RowBox[List["Floor", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]], RowBox[List["KroneckerDelta", "[", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "-", "Pi"]], "]"]]]]]]]]]]
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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> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <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> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⌋ </mo> </mrow> <mo> ⁢ </mo> <msub> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> KroneckerDelta </ci> </annotation-xml> </semantics> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> π </mi> </mrow> </msub> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </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 /> <cn type='integer'> -1 </cn> <apply> <ci> LogGamma </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <pi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <floor /> <apply> <plus /> <cn type='rational'> 3 <sep /> 4 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <imaginary /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> UnitStep </ci> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <floor /> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> KroneckerDelta </ci> <apply> <plus /> <apply> <arg /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
