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variants of this functions
PolyGamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Integral representations > Contour integral representations





http://functions.wolfram.com/06.15.07.0005.01









  


  








Input Form





PolyGamma[n, z] == (((-1)^(n + 1) n!)/(2 Pi I)) ContourIntegrate[(Gamma[s] Gamma[1 - s] Gamma[z - s]^(n + 1))/ Gamma[1 + z - s]^(n + 1)/(-1)^s, {s, \[ScriptCapitalL]}] /; Element[n, Integers] && n > 0








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "1"]]], " ", RowBox[List["n", "!"]]]], ")"]], " ", RowBox[List["ContourIntegrate", "[", 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["z", "-", "s"]], "]"]], RowBox[List["n", "+", "1"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "s"]]]]], SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "z", "-", "s"]], "]"]], RowBox[List["n", "+", "1"]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]


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





2001-10-29