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variants of this functions
EllipticPi




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticPi[n,m] > Integral representations > Contour integral representations





http://functions.wolfram.com/08.03.07.0004.01









  


  








Input Form





EllipticPi[n, m] == (1/(2 (2 Pi I)^2)) ContourIntegrate[ (Gamma[1/2 - s - t] Gamma[s] Gamma[1 - s] Gamma[t] Gamma[1/2 - t])/ Gamma[1 - s - t]/((-n)^s (-m)^t), {t, SuperStar[\[ScriptCapitalL]]}, {s, \[ScriptCapitalL]}]








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["ContourIntegrate", "[", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s", "-", "t"]], "]"]], " ", RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "s"]], "]"]], " ", RowBox[List["Gamma", "[", "t", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "t"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "n"]], ")"]], RowBox[List["-", "s"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "m"]], ")"]], RowBox[List["-", "t"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "s", "-", "t"]], "]"]]], ",", RowBox[List["{", RowBox[List["t", ",", SuperscriptBox["\[ScriptCapitalL]", "*"]]], "}"]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]], "2"]]]]]]]]


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





2001-10-29