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 EllipticPi

 http://functions.wolfram.com/08.03.06.0015.01

 Input Form

 EllipticPi[n, m] == (Floor[Arg[x - n]/(2 Pi)] ((Pi I)/Sqrt[(x - 1) ((x - m)/x)]) Sum[((((-1)^k Pochhammer[-(1/2), j] Pochhammer[1/2, i] Pochhammer[1/2, -i + k - j])/(i! (-i + k - j)! j!)) (-1 + x)^(i - k + j) (n - x)^k)/(x^j (-m + x)^i), {k, 0, Infinity}, {j, 0, k}, {i, 0, k - j}])/ E^(I Pi (Floor[(Pi - Arg[-m + n])/(2 Pi)] + Floor[(Pi - Arg[(x - m)/x] - Arg[1 + (n - x)/(x - m)])/(2 Pi)])) + (Pi/2) Sum[(Pochhammer[1/2, k]/k!) AppellF1[1/2 + k, 1 + k, 1/2, 1 + k, x, m] (n - x)^k, {k, 0, Infinity}] /; Element[x, Reals] && x > 1

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "\[Equal]", 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "j"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "j"]]]], "]"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], RowBox[List["i", "-", "k", "+", "j"]]], " ", SuperscriptBox["x", RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "+", "x"]], ")"]], RowBox[List["-", "i"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "k"]]]]]]]]]]], "+", RowBox[List[FractionBox["\[Pi]", "2"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], RowBox[List["k", "!"]]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", FractionBox["1", "2"], ",", RowBox[List["1", "+", "k"]], ",", "x", ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "k"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["x", ">", "1"]]]]]]]]

 MathML Form

 Π ( n m ) π 2 k = 0 ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] k ! F 1 AppellF1 ( k + 1 2 ; k + 1 , 1 2 ; k + 1 ; x , m ) ( n - x ) k + π ( x - 1 ) ( x - m ) x arg ( x - n ) 2 π - π ( - arg ( n - x x - m + 1 ) - arg ( x - m x ) + π 2 π + π - arg ( n - m ) 2 π ) k = 0 j = 0 k i = 0 k - j ( - 1 ) k ( - 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["1", "2"]]], ")"]], "j"], Pochhammer] ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] ( 1 2 ) - i - j + k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List[RowBox[List["-", "i"]], "-", "j", "+", "k"]]], Pochhammer] i ! ( - i - j + k ) ! j ! ( x - 1 ) i + j - k x - j ( x - m ) - i ( n - x ) k /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x > 1 Condition EllipticPi n m 2 -1 k 0 Pochhammer 1 2 k k -1 AppellF1 k 1 2 k 1 1 2 k 1 x m n -1 x k x -1 x -1 m x -1 1 2 -1 x -1 n 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 i 0 k -1 j j 0 k k 0 -1 k Pochhammer -1 1 2 j Pochhammer 1 2 i Pochhammer 1 2 -1 i -1 j k i -1 i -1 j k j -1 x -1 i j -1 k x -1 j x -1 m -1 i n -1 x k 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[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], RowBox[List["k", "-", "j"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "j"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "x"]], ")"]], RowBox[List["i", "-", "k", "+", "j"]]], " ", SuperscriptBox["x", RowBox[List["-", "j"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "m"]], "+", "x"]], ")"]], RowBox[List["-", "i"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "k"]]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "i"]], "+", "k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["j", "!"]]]]]]]]]]]]], SqrtBox[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["x", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["x", "-", "m"]], ")"]]]], "x"]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["AppellF1", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "k"]], ",", RowBox[List["1", "+", "k"]], ",", FractionBox["1", "2"], ",", RowBox[List["1", "+", "k"]], ",", "x", ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "x"]], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", ">", "1"]]]]]]]]]]

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

 2007-05-02