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http://functions.wolfram.com/06.14.06.0021.01
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PolyGamma[z + \[Epsilon]] \[Proportional]
PolyGamma[z] + Zeta[2, z] \[Epsilon] - Zeta[3, z] \[Epsilon]^2 +
O[\[Epsilon]^3] /; !(Element[z, Integers] && z <= 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["z", "+", "\[Epsilon]"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["PolyGamma", "[", "z", "]"]], "+", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["2", ",", "z"]], "]"]], "\[Epsilon]"]], " ", "-", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["3", ",", "z"]], "]"]], SuperscriptBox["\[Epsilon]", "2"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox["\[Epsilon]", "3"], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List["z", "\[Element]", "Integers"]], "\[And]", RowBox[List["z", "\[LessEqual]", "0"]]]], "]"]]]]]]
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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> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> ϵ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["2", Rule[Editable, True]], ",", TagBox["z", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> <mo> ⁢ </mo> <mi> ϵ </mi> </mrow> <mo> - </mo> <mrow> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> , </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", RowBox[List[TagBox["3", Rule[Editable, True]], ",", TagBox["z", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[List[ZetaDump`e1$, ZetaDump`e2$], Zeta[ZetaDump`e1$, ZetaDump`e2$]]]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mi> ϵ </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mi> ϵ </mi> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> z </mi> <mo> ≤ </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> z </ci> <ci> ϵ </ci> </apply> </apply> <apply> <plus /> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <ci> ϵ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> <ci> z </ci> </apply> <apply> <power /> <ci> ϵ </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> ϵ </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <not /> <apply> <and /> <apply> <in /> <ci> z </ci> <integers /> </apply> <apply> <leq /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List["z_", "+", "\[Epsilon]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", "z", "]"]], "+", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["2", ",", "z"]], "]"]], " ", "\[Epsilon]"]], "-", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["3", ",", "z"]], "]"]], " ", SuperscriptBox["\[Epsilon]", "2"]]], "+", SuperscriptBox[RowBox[List["O", "[", "\[Epsilon]", "]"]], "3"]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Element]", "Integers"]], "&&", RowBox[List["z", "\[LessEqual]", "0"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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