Arg[Gamma[x + I y]] == y PolyGamma[x] + Sum[y/(x + k) - ArcTan[y/(x + k)], {k, 0, Infinity}] /; Element[y, Reals] && x > 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Arg", "[", RowBox[List["Gamma", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["y", " ", RowBox[List["PolyGamma", "[", "x", "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["y", RowBox[List["x", "+", "k"]]], "-", RowBox[List["ArcTan", "[", FractionBox["y", RowBox[List["x", "+", "k"]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["y", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "0"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> y </mi> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> y </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> x </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> y </mi> <mrow> <mi> k </mi> <mo> + </mo> <mi> x </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> y </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arg /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> y </ci> <apply> <ci> PolyGamma </ci> <ci> x </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <plus /> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctan /> <apply> <times /> <ci> y </ci> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> x </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> y </ci> <reals /> </apply> <apply> <gt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Arg", "[", RowBox[List["Gamma", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["y", " ", RowBox[List["PolyGamma", "[", "x", "]"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List["(", RowBox[List[FractionBox["y", RowBox[List["x", "+", "k"]]], "-", RowBox[List["ArcTan", "[", FractionBox["y", RowBox[List["x", "+", "k"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["y", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "0"]]]]]]]]]]

2001-10-29