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variants of this functions
HarmonicNumber




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > HarmonicNumber[z] > Identities > Functional identities > Relations of special kind





http://functions.wolfram.com/06.16.17.0006.01









  


  








Input Form





HarmonicNumber[-z] == HarmonicNumber[z] - 1/z + Pi Cot[Pi z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["-", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "-", FractionBox["1", "z"], "+", " ", RowBox[List["\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msub> <mo> &#10869; </mo> <mrow> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mi> z </mi> </msub> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> cot </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HarmonicNumber </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <ci> HarmonicNumber </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <cot /> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["-", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["HarmonicNumber", "[", "z", "]"]], "-", FractionBox["1", "z"], "+", RowBox[List["\[Pi]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]]]]]]]]


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





2001-10-29