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 EllipticPi

 http://functions.wolfram.com/08.06.06.0051.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] EllipticPi[n, x, m] + ((1/Sqrt[m]) EllipticPi[n/m, 1/m] + (Pi I Sqrt[n])/(2 Sqrt[n - 1] Sqrt[n - m])) (1 - Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]]) + Exp[(-Pi) I Floor[Arg[x - z]/(2 Pi)]] ((1/(Sqrt[1 - m Sin[x]^2] (1 - n Sin[x]^2))) (z - x) - ((Sin[2 x] (-m - 2 n + 3 m n Sin[x]^2))/(4 (1 - m Sin[x]^2)^(3/2) (n Sin[x]^2 - 1)^2)) (z - x)^2 + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && Element[m, Reals] && Element[n, Reals] && 1 < m < n && ArcCsc[Sqrt[m]] + Pi u < x < Pi u + Pi/2 && Element[u, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "x", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SqrtBox["m"]], RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", SqrtBox["n"]]], RowBox[List["2", SqrtBox[RowBox[List["n", "-", "1"]]], " ", SqrtBox[RowBox[List["n", "-", "m"]]]]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Exp", "[", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]]]]], RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[RowBox[List["Sin", "[", RowBox[List["2", "x"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["3", " ", "m", " ", "n", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]], "-", "1"]], ")"]], "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["m", "\[Element]", "Reals"]], "\[And]", RowBox[List["n", "\[Element]", "Reals"]], "\[And]", RowBox[List["1", "<", "m", "<", "n"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[RowBox[List["\[Pi]", " ", "u"]], "+", FractionBox["\[Pi]", "2"]]]]], "\[And]", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]

 MathML Form

 Π ( n ; z m ) ( 1 m Π ( n m 1 m ) + π n 2 n - 1 n - m ) ( 1 - - π arg ( x - z ) 2 π ) + Π ( n ; x m ) - π arg ( x - z ) 2 π + - π arg ( x - z ) 2 π ( z - x 1 - m sin 2 ( x ) ( 1 - n sin 2 ( x ) ) - sin ( 2 x ) ( 3 m n sin 2 ( x ) - m - 2 n ) 4 ( 1 - m sin 2 ( x ) ) 3 / 2 ( n sin 2 ( x ) - 1 ) 2 ( z - x ) 2 + ) /; ( z "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] n TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] 1 < m < n π u + csc - 1 ( m ) < x < π u + π 2 u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional EllipticPi n z m 1 m 1 2 -1 EllipticPi n m -1 1 m -1 n 1 2 2 n -1 1 2 n -1 m 1 2 -1 1 -1 -1 x -1 z 2 -1 EllipticPi n x m -1 x -1 z 2 -1 -1 x -1 z 2 -1 z -1 x 1 -1 m x 2 1 2 1 -1 n x 2 -1 -1 2 x 3 m n x 2 -1 m -1 2 n 4 1 -1 m x 2 3 2 n x 2 -1 2 -1 z -1 x 2 Rule z x x m n 1 m n u m 1 2 x u 2 -1 u [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "x", ",", "m"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]], SqrtBox["m"]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SqrtBox["n"]]], RowBox[List["2", " ", SqrtBox[RowBox[List["n", "-", "1"]]], " ", SqrtBox[RowBox[List["n", "-", "m"]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["z", "-", "x"]], RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Sin", "[", RowBox[List["2", " ", "x"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "-", RowBox[List["2", " ", "n"]], "+", RowBox[List["3", " ", "m", " ", "n", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]]]], ")"]], RowBox[List["3", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["n", " ", SuperscriptBox[RowBox[List["Sin", "[", "x", "]"]], "2"]]], "-", "1"]], ")"]], "2"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["m", "\[Element]", "Reals"]], "&&", RowBox[List["n", "\[Element]", "Reals"]], "&&", RowBox[List["1", "<", "m", "<", "n"]], "&&", RowBox[List[RowBox[List[RowBox[List["ArcCsc", "[", SqrtBox["m"], "]"]], "+", RowBox[List["\[Pi]", " ", "u"]]]], "<", "x", "<", RowBox[List[RowBox[List["\[Pi]", " ", "u"]], "+", FractionBox["\[Pi]", "2"]]]]], "&&", RowBox[List["u", "\[Element]", "Integers"]]]]]]]]]]

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

 2007-05-02