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variants of this functions
HarmonicNumber






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > HarmonicNumber[z] > Series representations > Generalized power series > Expansions at z==z0/;z0!=-n > For the function itself





http://functions.wolfram.com/06.16.06.0026.01









  


  










Input Form





HarmonicNumber[z] \[Proportional] HarmonicNumber[Subscript[z, 0]] + Zeta[2, Subscript[z, 0] + 1] (z - Subscript[z, 0]) - Zeta[3, Subscript[z, 0] + 1] (z - Subscript[z, 0])^2 + Zeta[4, Subscript[z, 0] + 1] (z - Subscript[z, 0])^3 + O[(z - Subscript[z, 0])^4] /; !(Element[Subscript[z, 0], Integers] && Subscript[z, 0] < 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["HarmonicNumber", "[", SubscriptBox["z", "0"], "]"]], "+", " ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["2", ",", RowBox[List[SubscriptBox["z", "0"], "+", "1"]]]], "]"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "-", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["3", ",", RowBox[List[SubscriptBox["z", "0"], "+", "1"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], " ", "+", " ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["4", ",", RowBox[List[SubscriptBox["z", "0"], "+", "1"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "4"], "]"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["z", "0"], "\[Element]", "Integers"]], "\[And]", RowBox[List[SubscriptBox["z", "0"], "<", "0"]]]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> &#8733; </mo> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </msub> <mo> + </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;2&quot;, Rule[Editable, True]], &quot;,&quot;, RowBox[List[TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;+&quot;, &quot;1&quot;]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> , </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;3&quot;, Rule[Editable, True]], &quot;,&quot;, RowBox[List[TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;+&quot;, &quot;1&quot;]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> , </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;4&quot;, Rule[Editable, True]], &quot;,&quot;, RowBox[List[TagBox[SubscriptBox[&quot;z&quot;, &quot;0&quot;], Rule[Editable, True]], &quot;+&quot;, &quot;1&quot;]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> &#172; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> &lt; </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> HarmonicNumber </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 4 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <not /> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <integers /> </apply> <apply> <lt /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", SubscriptBox["zz", "0"], "]"]], "+", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["2", ",", RowBox[List[SubscriptBox["zz", "0"], "+", "1"]]]], "]"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "-", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["3", ",", RowBox[List[SubscriptBox["zz", "0"], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["4", ",", RowBox[List[SubscriptBox["zz", "0"], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "3"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "4"]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["zz", "0"], "\[Element]", "Integers"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "<", "0"]]]], ")"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02





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