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variants of this functions
HarmonicNumber






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > HarmonicNumber[z,r] > Integral representations > Contour integral representations





http://functions.wolfram.com/06.17.07.0004.01









  


  










Input Form





HarmonicNumber[z, r] == (1/(2 Pi I)) Integrate[(Gamma[s] Gamma[1 - s] Gamma[1 - s]^r)/ Gamma[2 - s]^r/(-1)^s, {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] - (1/(2 Pi I)) Integrate[(Gamma[s] Gamma[1 - s] Gamma[1 + z - s]^r)/Gamma[2 + z - s]^r/ (-1)^s, {s, \[Gamma] - I Infinity, \[Gamma] + I Infinity}] /; 0 < \[Gamma] < 1 && Element[r - 1, Integers] && r - 1 > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HarmonicNumber", "[", RowBox[List["z", ",", "r"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], "r"]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "-", "s"]], "]"]], "r"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "z", "-", "s"]], "]"]], "r"]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "+", "z", "-", "s"]], "]"]], "r"]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "s"]]], RowBox[List["\[DifferentialD]", "s"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]", "<", "1"]], "\[And]", RowBox[List[RowBox[List["r", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["r", "-", "1"]], ">", "0"]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HarmonicNumber", "[", RowBox[List["z_", ",", "r_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], "r"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "s"]]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "-", "s"]], "]"]], "r"]], RowBox[List["\[DifferentialD]", "s"]]]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "-", FractionBox[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["\[Gamma]", "-", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]], RowBox[List["\[Gamma]", "+", RowBox[List["\[ImaginaryI]", " ", "\[Infinity]"]]]]], RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "z", "-", "s"]], "]"]], "r"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "s"]]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["2", "+", "z", "-", "s"]], "]"]], "r"]], RowBox[List["\[DifferentialD]", "s"]]]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]], "/;", RowBox[List[RowBox[List["0", "<", "\[Gamma]", "<", "1"]], "&&", RowBox[List[RowBox[List["r", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["r", "-", "1"]], ">", "0"]]]]]]]]]]










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





2001-10-29





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