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http://functions.wolfram.com/06.17.06.0004.01
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HarmonicNumber[z, r] \[Proportional] r Zeta[r + 1] z (1 + O[z]) /; (z -> 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "\[Proportional]", RowBox[List["r", " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "1"]], "]"]], "z", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "z", "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "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> <msubsup> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> <mrow> <mo> ( </mo> <mi> r </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ∝ </mo> <mrow> <mi> r </mi> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> r </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["r", "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> <ci> r </ci> </apply> <apply> <times /> <ci> r </ci> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> r </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["r", " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "1"]], "]"]], " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["O", "[", "z", "]"]]]], ")"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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