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http://functions.wolfram.com/06.14.19.0001.01
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Re[PolyGamma[x + I y]] == -EulerGamma +
RootSum[(1 + #1) (x^2 + y^2 + 2 x #1 + #1^2) & ,
-((PolyGamma[-#1] (x^2 + y^2 + x (-1 + #1) - #1))/
(x^2 + y^2 + #1 (2 + 3 #1) + x (2 + 4 #1))) & ]
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Cell[BoxData[RowBox[List[RowBox[List["Re", "[", RowBox[List["PolyGamma", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List["RootSum", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "#1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", "x", " ", "#1"]], "+", SuperscriptBox["#1", "2"]]], ")"]]]], "&"]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["-", "#1"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["x", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "#1"]], ")"]]]], "-", "#1"]], ")"]]]], RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["#1", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["3", " ", "#1"]]]], ")"]]]], "+", RowBox[List["x", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "#1"]]]], ")"]]]]]]]]], "&"]]]], "]"]]]]]]]]
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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> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> RootSum </mi> <mo> [ </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> #1 </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> #1 </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> #1 </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> & </mo> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 0 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mi> #1 </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> #1 </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mi> #1 </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> #1 </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mi> #1 </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> #1 </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mfrac> </mrow> <mo> & </mo> </mrow> </mrow> <mo> ] </mo> </mrow> <mo> - </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <real /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> RootSum </ci> <apply> <ci> Function </ci> <apply> <times /> <apply> <plus /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Function </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 0 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Slot </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", RowBox[List["PolyGamma", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", "EulerGamma"]], "+", RowBox[List["RootSum", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "#1"]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["2", " ", "x", " ", "#1"]], "+", SuperscriptBox["#1", "2"]]], ")"]]]], "&"]], ",", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["-", "#1"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["x", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "#1"]], ")"]]]], "-", "#1"]], ")"]]]], RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"], "+", RowBox[List["#1", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["3", " ", "#1"]]]], ")"]]]], "+", RowBox[List["x", " ", RowBox[List["(", RowBox[List["2", "+", RowBox[List["4", " ", "#1"]]]], ")"]]]]]]]]], "&"]]]], "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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