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variants of this functions
PolyGamma




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.14.07.0005.01









  


  








Input Form





PolyGamma[z] == (-(1/(Gamma[1 - z] 2 Pi I))) ContourIntegrate[ (Gamma[s] Gamma[1 - s] Gamma[1 - s] Gamma[2 - z - s])/ (Gamma[2 - s] Gamma[2 - s])/(-1)^s, {s, \[ScriptCapitalL]}] - EulerGamma








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["PolyGamma", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "z"]], "]"]], "2", "\[Pi]", " ", "\[ImaginaryI]"]]]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "z", "-", "s"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", "-", "s"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["2", "-", "s"]], "]"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], "-", "EulerGamma"]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "z", "-", "s"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", "s"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "s"]], "]"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "z"]], "]"]], " ", "2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]]]], "-", "EulerGamma"]]]]]]


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





2001-10-29