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http://functions.wolfram.com/04.13.23.0021.01
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Sum[(((-1)^k BernoulliB[2 k])/(k Gamma[2 k - \[Alpha]])) z^(2 k),
{k, 1, Infinity}] == -2 (-(z/(2 Pi)))^(1 + \[Alpha])
PolyGamma[\[Alpha], -(z/(2 Pi))] - 2 (z/(2 Pi))^(1 + \[Alpha])
PolyGamma[\[Alpha], z/(2 Pi)] - (2/Gamma[-\[Alpha]])
(-2 EulerGamma + Log[-z] + Log[z] - 2 PolyGamma[-\[Alpha]] - 2 Log[2 Pi])
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", "k"]], "]"]]]], RowBox[List["k", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k"]], "-", "\[Alpha]"]], "]"]]]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", RowBox[List["2", "\[Pi]", " "]]]]], ")"]], RowBox[List["1", "+", "\[Alpha]"]]], RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox["z", RowBox[List["2", "\[Pi]"]]]]]]], "]"]]]], "-", " ", RowBox[List["2", SuperscriptBox[RowBox[List["(", FractionBox["z", RowBox[List["2", "\[Pi]"]]], ")"]], RowBox[List["1", "+", "\[Alpha]"]]], RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox["z", RowBox[List["2", "\[Pi]"]]]]], "]"]]]], "-", RowBox[List[FractionBox["2", RowBox[List["Gamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "EulerGamma"]], "+", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]]], "-", RowBox[List["2", RowBox[List["Log", "[", RowBox[List["2", "\[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> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> ⁢ </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msub> </mrow> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> </msup> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mtext> </mtext> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> α </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> α </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> α </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <ci> BernoulliB </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ci> PolyGamma </ci> <ci> α </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ln /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> α </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <eulergamma /> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> α </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> α </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k_"], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "k_"]], "]"]]]], ")"]], " ", SuperscriptBox["z_", RowBox[List["2", " ", "k_"]]]]], RowBox[List["k_", " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["2", " ", "k_"]], "-", "\[Alpha]_"]], "]"]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", RowBox[List["2", " ", "\[Pi]"]]]]], ")"]], RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox["z", RowBox[List["2", " ", "\[Pi]"]]]]]]], "]"]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", FractionBox["z", RowBox[List["2", " ", "\[Pi]"]]], ")"]], RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox["z", RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "EulerGamma"]], "+", RowBox[List["Log", "[", RowBox[List["-", "z"]], "]"]], "+", RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Log", "[", RowBox[List["2", " ", "\[Pi]"]], "]"]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
