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 EllipticPi

 http://functions.wolfram.com/08.03.06.0013.01

 Input Form

 EllipticPi[n, m] \[Proportional] (Floor[Arg[x - n]/(2 Pi)] ((Pi I)/Sqrt[(x - 1) ((x - m)/x)]) (1 + ((-m + x^2)/(2 (m - x) (-1 + x) x)) (n - x) + ((2 m (2 - 5 x) x + 3 x^4 + m^2 (-1 + 4 x))/(8 (m - x)^2 (-1 + x)^2 x^2)) (n - x)^2 + \[Ellipsis]))/ E^(I Pi (Floor[(Pi - Arg[-m + n])/(2 Pi)] + Floor[(Pi - Arg[(x - m)/x] - Arg[1 + (n - x)/(x - m)])/(2 Pi)])) + EllipticPi[x, m] + (Pi/4) AppellF1[3/2, 2, 1/2, 2, x, m] (n - x) + ((3 Pi)/16) AppellF1[5/2, 3, 1/2, 3, x, m] (n - x)^2 + \[Ellipsis] /; (n -> x) && Element[x, Reals] && x > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "n"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], SqrtBox[RowBox[List[RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]], FractionBox[RowBox[List["x", "-", "m"]], "x"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "-", "m"]], "x"], "]"]], "-", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "x"]], RowBox[List["x", "-", "m"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["-", "m"]], "+", SuperscriptBox["x", "2"]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "x"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], " ", "x"]]], RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", "x"]]]], ")"]], " ", "x"]], "+", RowBox[List["3", " ", SuperscriptBox["x", "4"]]], "+", RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "x"]]]], ")"]]]]]], RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "x"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], " ", SuperscriptBox["x", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["EllipticPi", "[", RowBox[List["x", ",", "m"]], "]"]], "+", RowBox[List[FractionBox["\[Pi]", "4"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", "2", ",", FractionBox["1", "2"], ",", "2", ",", "x", ",", "m"]], "]"]], RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["3", "\[Pi]"]], "16"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", "3", ",", FractionBox["1", "2"], ",", "3", ",", "x", ",", "m"]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "x"]], ")"]], "\[And]", RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "1"]]]]]]]]

 MathML Form

 Π ( n m ) arg ( x - n ) 2 π π ( x - 1 ) ( x - m ) x - π ( π - arg ( n - x x - m + 1 ) - arg ( x - m x ) 2 π + π - arg ( n - m ) 2 π ) ( 1 + ( x 2 - m ) ( n - x ) 2 ( m - x ) ( x - 1 ) x + ( 3 x 4 + 2 m ( 2 - 5 x ) x + m 2 ( 4 x - 1 ) ) ( n - x ) 2 8 ( m - x ) 2 ( x - 1 ) 2 x 2 + ) + Π ( x m ) + π 4 F 1 AppellF1 ( 3 2 ; 2 , 1 2 ; 2 ; x , m ) ( n - x ) + 3 π 16 F 1 AppellF1 ( 5 2 ; 3 , 1 2 ; 3 ; x , m ) ( n - x ) 2 + /; ( n "\[Rule]" x ) x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x > 1 Condition Proportional EllipticPi n m x -1 n 2 -1 x -1 x -1 m x -1 1 2 -1 -1 -1 n -1 x x -1 m -1 1 -1 x -1 m x -1 2 -1 -1 n -1 m 2 -1 1 x 2 -1 m n -1 x 2 m -1 x x -1 x -1 3 x 4 2 m 2 -1 5 x x m 2 4 x -1 n -1 x 2 8 m -1 x 2 x -1 2 x 2 -1 Π VerticalSeparator x m 4 -1 AppellF1 3 2 2 1 2 2 x m n -1 x 3 16 -1 AppellF1 5 2 3 1 2 3 x m n -1 x 2 Rule n x x x 1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["x", "-", "n"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List[RowBox[List["-", "m"]], "+", "n"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["x", "-", "m"]], "x"], "]"]], "-", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox[RowBox[List["n", "-", "x"]], RowBox[List["x", "-", "m"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "+", SuperscriptBox["x", "2"]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "x"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], " ", "x"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "m", " ", RowBox[List["(", RowBox[List["2", "-", RowBox[List["5", " ", "x"]]]], ")"]], " ", "x"]], "+", RowBox[List["3", " ", SuperscriptBox["x", "4"]]], "+", RowBox[List[SuperscriptBox["m", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", "x"]]]], ")"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "2"]]], RowBox[List["8", " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "x"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], "2"], " ", SuperscriptBox["x", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["x", "-", "m"]], ")"]]]], "x"]]], "+", RowBox[List["EllipticPi", "[", RowBox[List["x", ",", "m"]], "]"]], "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", "2", ",", FractionBox["1", "2"], ",", "2", ",", "x", ",", "m"]], "]"]], " ", RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]]]], "+", RowBox[List[FractionBox["1", "16"], " ", RowBox[List["(", RowBox[List["3", " ", "\[Pi]"]], ")"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", "3", ",", FractionBox["1", "2"], ",", "3", ",", "x", ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "1"]]]]]]]]]]

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

 2007-05-02