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StieltjesGamma




Mathematica Notation

Traditional Notation









Zeta Functions and Polylogarithms > StieltjesGamma[n] > Integral representations > Contour integral representations





http://functions.wolfram.com/10.05.07.0001.01









  


  








Input Form





StieltjesGamma[n] == (((-1)^n n!)/(2 Pi I)) ContourIntegrate[ (Zeta[s] - 1/(s - 1)) (1/(s - 1)^(n + 1)), {s, Abs[s - 1] == 1}] /; Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["StieltjesGamma", "[", "n", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]]]], RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]]], RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Zeta", "[", "s", "]"]], "-", FractionBox["1", RowBox[List["s", "-", "1"]]]]], ")"]], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], RowBox[List["n", "+", "1"]]]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["Abs", "[", RowBox[List["s", "-", "1"]], "]"]], "\[Equal]", "1"]]]], "}"]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["StieltjesGamma", "[", "n_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]]]], ")"]], " ", RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["Zeta", "[", "s", "]"]], "-", FractionBox["1", RowBox[List["s", "-", "1"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["s", "-", "1"]], ")"]], RowBox[List["n", "+", "1"]]]], ",", RowBox[List["{", RowBox[List["s", ",", RowBox[List[RowBox[List["Abs", "[", RowBox[List["s", "-", "1"]], "]"]], "\[Equal]", "1"]]]], "}"]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29