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 HarmonicNumber

 http://functions.wolfram.com/06.17.06.0020.01

 Input Form

 HarmonicNumber[z, n] \[Proportional] Zeta[n] - (n + 2 z + 1)/(2 (z + 1)^n (n - 1)) - (1/(n - 1)!) Sum[((2 k + n - 2)!/((2 k)! (z + 1)^(2 k + n - 1))) BernoulliB[2 k], {k, 1, Infinity}] + Floor[Abs[Arg[z + 1]]/Pi] (((I Pi)^n 2^(n - 1))/(n - 1)!) (I Cot[Pi z] - 1) Sum[((-1)^k k! StirlingS2[n - 1, k] (I Cot[Pi z] + 1)^k)/2^k, {k, 0, n - 1}] /; (Abs[z] -> Infinity) && Element[n, Integers] && n > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "n"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List["Zeta", "[", "n", "]"]], "-", FractionBox[RowBox[List["n", "+", RowBox[List["2", " ", "z"]], "+", "1"]], RowBox[List["2", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "n"], RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]]], "-", RowBox[List[FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "n", "-", "2"]], ")"]], "!"]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", "k"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[RowBox[List["2", "k"]], "+", "n", "-", "1"]]]]]], RowBox[List["BernoulliB", "[", RowBox[List["2", "k"]], "]"]]]]]]]], "+", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["z", "+", "1"]], "]"]], "]"]], "\[Pi]"], "]"]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["n", "-", "1"]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["k", "!"]], " ", RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], "+", "1"]], ")"]], "k"]]], SuperscriptBox["2", "k"]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "1"]]]]]]]]

 MathML Form

 H HarmonicNumber z ( n ) ζ ( n ) TagBox[RowBox[List["\[Zeta]", "(", TagBox["n", Zeta, Rule[Editable, True], Rule[Selectable, True]], ")"]], InterpretTemplate[Function[BoxForm`e\$, Zeta[BoxForm`e\$]]], Rule[Editable, False], Rule[Selectable, False]] - n + 2 z + 1 2 ( z + 1 ) n ( n - 1 ) + "\[LeftBracketingBar]" arg ( z + 1 ) "\[RightBracketingBar]" π ( cot ( π z ) - 1 ) ( π ) n 2 n - 1 ( n - 1 ) ! k = 0 n - 1 ( - 1 ) k k ! 2 k 𝒮 TagBox["\[ScriptCapitalS]", StirlingS2] n - 1 ( k ) ( cot ( π z ) + 1 ) k - 1 ( n - 1 ) ! k = 1 ( 2 k + n - 2 ) ! B TagBox["B", BernoulliB] 2 k ( 2 k ) ! ( z + 1 ) 2 k + n - 1 /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) n TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] n > 1 Condition Proportional HarmonicNumber z n Zeta n -1 n 2 z 1 2 z 1 n n -1 -1 z 1 -1 z -1 n 2 n -1 n -1 -1 k 0 n -1 -1 k k 2 k -1 StirlingS2 n -1 k z 1 k -1 1 n -1 -1 k 1 2 k n -2 BernoulliB 2 k 2 k z 1 2 k n -1 -1 Rule z n n 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "n_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Zeta", "[", "n", "]"]], "-", FractionBox[RowBox[List["n", "+", RowBox[List["2", " ", "z"]], "+", "1"]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]]], "-", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "n", "-", "2"]], ")"]], "!"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["2", " ", "k"]], "]"]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "k"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "n", "-", "1"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["z", "+", "1"]], "]"]], "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["n", "-", "1"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], "-", "1"]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["k", "!"]], " ", RowBox[List["StirlingS2", "[", RowBox[List[RowBox[List["n", "-", "1"]], ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]]]], "+", "1"]], ")"]], "k"]]], SuperscriptBox["2", "k"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "1"]]]]]]]]]]

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

 2007-05-02