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 EllipticPi

 http://functions.wolfram.com/08.06.06.0102.01

 Input Form

 EllipticPi[n, z, m] == EllipticF[z, m] - (-1)^Round[Re[z]/Pi] ((n Sqrt[(-m) Sin[z]^2])/(m Sin[z])) ((1/(Sqrt[-1 + n] Sqrt[n])) (ArcTanh[(Cos[z] Sqrt[n])/Sqrt[-1 + n]] - ArcTanh[Sqrt[n]/Sqrt[-1 + n]]) + ((1 - Sqrt[(m - n)/m])/(Sqrt[(m - n)/m] (-1 + n))) (-ArcTanh[Sqrt[n]/Sqrt[-1 + n]] + ArcTanh[(Sqrt[n] Cos[z])/Sqrt[-1 + n]] Cos[z]) + (1/(2 n m)) Sum[(Pochhammer[3/2, k]/(m^k k!)) (HypergeometricPFQ[{{k + 2}, {1/2, 1}, {1}}, {{2}, {3/2}, {}}, 1, (n - 1)/n] - Cos[z] HypergeometricPFQ[{{k + 2}, {1/2, 1}, {1}}, {{2}, {3/2}, {}}, Cos[z]^2, (n - 1)/n]), {k, 0, Infinity}]) + 2 Round[Re[z]/Pi] (EllipticPi[n, m] - EllipticK[m])

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "-", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], FractionBox[RowBox[List["n", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], RowBox[List["m", " ", RowBox[List["Sin", "[", "z", "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SqrtBox["n"]]]], RowBox[List["(", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["Cos", "[", "z", "]"]], SqrtBox["n"]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " "]]], "]"]], "-", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox["n"], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " "]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["(", RowBox[List["1", "-", SqrtBox[FractionBox[RowBox[List["m", "-", "n"]], "m"]]]], ")"]], RowBox[List[SqrtBox[FractionBox[RowBox[List["m", "-", "n"]], "m"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox["n"], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]]]], "+", " ", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox["n"], RowBox[List["Cos", "[", "z", "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], RowBox[List["Cos", "[", "z", "]"]]]]]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", "n", " ", "m"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["m", RowBox[List["-", "k"]]]]], RowBox[List["k", "!"]]], RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "2", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "1", ",", FractionBox[RowBox[List["n", "-", "1"]], "n"]]], "]"]], "-", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "2", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"], ",", FractionBox[RowBox[List["n", "-", "1"]], "n"]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "-", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]]]]]]]]]]

 MathML Form

 Π ( n ; z m ) F ( z m ) + 2 ( Π ( n m ) - K ( m ) ) Re ( z ) π - ( - 1 ) Re ( z ) π n - m sin 2 ( z ) m sin ( z ) ( tanh - 1 ( cos ( z ) n n - 1 ) - tanh - 1 ( n n - 1 ) n - 1 n + ( 1 - m - n m ) m - n m ( n - 1 ) ( tanh - 1 ( n cos ( z ) n - 1 ) cos ( z ) - tanh - 1 ( n n - 1 ) ) + 1 2 n m k = 0 ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] m - k k ! ( F 1 1 0 1 2 1 ( k + 2 ; 1 2 , 1 ; 1 ; 2 ; 3 2 ; ; 1 , n - 1 n ) - cos ( z ) F 1 1 0 1 2 1 ( k + 2 ; 1 2 , 1 ; 1 ; 2 ; 3 2 ; ; cos 2 ( z ) , n - 1 n ) ) ) Π ( n ; z m ) F ( z m ) + 2 ( Π ( n m ) - K ( m ) ) Re ( z ) π - ( - 1 ) Re ( z ) π n - m sin 2 ( z ) m sin ( z ) ( tanh - 1 ( cos ( z ) n n - 1 ) - tanh - 1 ( n n - 1 ) n - 1 n + ( 1 - m - n m ) m - n m ( n - 1 ) ( tanh - 1 ( n cos ( z ) n - 1 ) cos ( z ) - tanh - 1 ( n n - 1 ) ) + 1 2 n m k = 0 ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] m - k k ! ( F 1 1 0 1 2 1 ( k + 2 ; 1 2 , 1 ; 1 ; 2 ; 3 2 ; ; 1 , n - 1 n ) - cos ( z ) F 1 1 0 1 2 1 ( k + 2 ; 1 2 , 1 ; 1 ; 2 ; 3 2 ; ; cos 2 ( z ) , n - 1 n ) ) ) [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticPi", "[", RowBox[List["n_", ",", "z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List["n", " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", SqrtBox["n"]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], "-", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox["n"], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]]]], RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SqrtBox["n"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SqrtBox[FractionBox[RowBox[List["m", "-", "n"]], "m"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["ArcTanh", "[", FractionBox[SqrtBox["n"], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]]]], "+", RowBox[List[RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox["n"], " ", RowBox[List["Cos", "[", "z", "]"]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], " ", RowBox[List["Cos", "[", "z", "]"]]]]]], ")"]]]], RowBox[List[SqrtBox[FractionBox[RowBox[List["m", "-", "n"]], "m"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]]]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]], " ", SuperscriptBox["m", RowBox[List["-", "k"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "2", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", "1", ",", FractionBox[RowBox[List["n", "-", "1"]], "n"]]], "]"]], "-", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["k", "+", "2"]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]], ",", RowBox[List["{", "1", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", "2", "}"]], ",", RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], "2"], ",", FractionBox[RowBox[List["n", "-", "1"]], "n"]]], "]"]]]]]], ")"]]]], RowBox[List["k", "!"]]]]], RowBox[List["2", " ", "n", " ", "m"]]]]], ")"]]]], RowBox[List["m", " ", RowBox[List["Sin", "[", "z", "]"]]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "-", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]]]]]]]]]]

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

 2007-05-02