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 EllipticPi

 http://functions.wolfram.com/08.06.06.0088.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] ((Sqrt[n] ArcTanh[Sqrt[n] Sin[z]] - ArcTanh[Sin[z]])/ (n - 1) - (1/(4 (-1 + n)^2)) ((1 + n) ArcTanh[Sin[z]] - 2 Sqrt[n] ArcTanh[Sqrt[n] Sin[z]] + (-1 + n) Sec[z] Tan[z]) (m - 1) + (3/(64 (-1 + n)^3)) ((-3 - 6 n + n^2) ArcTanh[Sin[z]] + 8 Sqrt[n] ArcTanh[Sqrt[n] Sin[z]] + (1/2) (-1 + n) (-1 - 3 n + (-5 + n) Cos[2 z]) Sec[z]^3 Tan[z]) (m - 1)^2 + \[Ellipsis]) - Round[Re[z]/Pi] (Log[1 - m] (1/(1 - n)) (1 + ((n + 1) (m - 1))/(4 (n - 1)) - (3 (-3 - 6 n + n^2) (m - 1)^2)/(64 (n - 1)^2) + \[Ellipsis]) - (-4 Log[2] + Sqrt[n] (-Log[1 - Sqrt[n]] + Log[1 + Sqrt[n]]))/(-1 + n) + ((-1 + 2 Log[2] + 2 n Log[2] + Sqrt[n] (Log[1 - Sqrt[n]] - Log[1 + Sqrt[n]])) (m - 1))/(2 (-1 + n)^2) - (1/(64 (-1 + n)^3)) ((21 + 12 n - 5 n^2 + 12 (-3 + (-6 + n) n) Log[2] + 24 Sqrt[n] (-Log[1 - Sqrt[n]] + Log[1 + Sqrt[n]])) (m - 1)^2) + \[Ellipsis]) /; (m -> 1) && !Element[(Pi + 2 Re[z])/(4 Pi), Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "z", ",", "m"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[SqrtBox["n"], " ", RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox["n"], " ", RowBox[List["Sin", "[", "z", "]"]]]], "]"]]]], "-", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], RowBox[List["n", "-", "1"]]], "-", RowBox[List[FractionBox["1", RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ")"]], "2"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox["n"], " ", 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RowBox[List[RowBox[List["-", RowBox[List["Log", "[", RowBox[List["1", "-", SqrtBox["n"]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["1", "+", SqrtBox["n"]]], "]"]]]], ")"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], "2"]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["m", "\[Rule]", "1"]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]], "]"]]]]]]]]

 MathML Form

 Π ( n ; z m ) ( - 1 ) Re ( z ) π ( n tanh - 1 ( n sin ( z ) ) - tanh - 1 ( sin ( z ) ) n - 1 - ( n + 1 ) tanh - 1 ( sin ( z ) ) - 2 n tanh - 1 ( n sin ( z ) ) + ( n - 1 ) sec ( z ) tan ( z ) 4 ( n - 1 ) 2 ( m - 1 ) + 3 64 ( n - 1 ) 3 ( 1 2 ( n - 1 ) ( - 3 n + ( n - 5 ) cos ( 2 z ) - 1 ) tan ( z ) sec 3 ( z ) + ( n 2 - 6 n - 3 ) tanh - 1 ( sin ( z ) ) + 8 n tanh - 1 ( n sin ( z ) ) ) ( m - 1 ) 2 + ) - Re ( z ) π ( log ( 1 - m ) 1 - n ( 1 + ( n + 1 ) ( m - 1 ) 4 ( n - 1 ) - 3 ( n 2 - 6 n - 3 ) ( m - 1 ) 2 64 ( n - 1 ) 2 + ) - n ( log ( n + 1 ) - log ( 1 - n ) ) - 4 log ( 2 ) n - 1 + ( 2 n log ( 2 ) + 2 log ( 2 ) + n ( log ( 1 - n ) - log ( n + 1 ) ) - 1 ) ( m - 1 ) 2 ( n - 1 ) 2 - 1 64 ( n - 1 ) 3 ( - 5 n 2 + 12 n + 24 ( log ( n + 1 ) - log ( 1 - n ) ) n + 12 ( ( n - 6 ) n - 3 ) log ( 2 ) + 21 ) ( m - 1 ) 2 + ) /; ( m "\[Rule]" 1 ) ¬ 2 Re ( z ) + π 4 π TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] FormBox RowBox RowBox RowBox Π ( RowBox n ; RowBox z m ) RowBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) ErrorBox RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox FractionBox RowBox RowBox SqrtBox n RowBox SuperscriptBox tanh RowBox - 1 ( RowBox SqrtBox n RowBox sin ( z ) ) - RowBox SuperscriptBox tanh RowBox - 1 ( RowBox sin ( z ) ) RowBox n - 1 - RowBox FractionBox RowBox RowBox RowBox ( RowBox n + 1 ) RowBox SuperscriptBox tanh RowBox - 1 ( RowBox sin ( z ) ) - RowBox 2 SqrtBox n RowBox SuperscriptBox tanh RowBox - 1 ( RowBox SqrtBox n RowBox sin ( z ) ) + RowBox RowBox ( RowBox n - 1 ) RowBox sec ( z ) RowBox tan ( z ) RowBox 4 SuperscriptBox RowBox ( RowBox n - 1 ) 2 RowBox ( RowBox m - 1 ) + RowBox FractionBox 3 RowBox 64 SuperscriptBox RowBox ( RowBox n - 1 ) 3 RowBox ( RowBox RowBox FractionBox 1 2 RowBox ( RowBox n - 1 ) RowBox ( RowBox RowBox RowBox - 3 n + RowBox RowBox ( RowBox n - 5 ) RowBox cos ( RowBox 2 z ) - 1 ) RowBox tan ( z ) RowBox SuperscriptBox sec 3 ( z ) + RowBox RowBox ( RowBox SuperscriptBox n 2 - RowBox 6 n - 3 ) RowBox SuperscriptBox tanh RowBox - 1 ( RowBox sin ( z ) ) + RowBox 8 SqrtBox n RowBox SuperscriptBox tanh RowBox - 1 ( RowBox SqrtBox n RowBox sin ( z ) ) ) SuperscriptBox RowBox ( RowBox m - 1 ) 2 + ) - RowBox RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox RowBox FractionBox RowBox log ( RowBox 1 - m ) RowBox 1 - n RowBox ( RowBox 1 + FractionBox RowBox RowBox ( RowBox n + 1 ) RowBox ( RowBox m - 1 ) RowBox 4 RowBox ( RowBox n - 1 ) - FractionBox RowBox 3 RowBox ( RowBox SuperscriptBox n 2 - RowBox 6 n - 3 ) SuperscriptBox RowBox ( RowBox m - 1 ) 2 RowBox 64 SuperscriptBox RowBox ( RowBox n - 1 ) 2 + ) - FractionBox RowBox RowBox SqrtBox n RowBox ( RowBox RowBox log ( RowBox SqrtBox n + 1 ) - RowBox log ( RowBox 1 - SqrtBox n ) ) - RowBox 4 RowBox log ( 2 ) RowBox n - 1 + FractionBox RowBox RowBox ( RowBox RowBox 2 n RowBox log ( 2 ) + RowBox 2 RowBox log ( 2 ) + RowBox SqrtBox n RowBox ( RowBox RowBox log ( RowBox 1 - SqrtBox n ) - RowBox log ( RowBox SqrtBox n + 1 ) ) - 1 ) RowBox ( RowBox m - 1 ) RowBox 2 SuperscriptBox RowBox ( RowBox n - 1 ) 2 - RowBox FractionBox 1 RowBox 64 SuperscriptBox RowBox ( RowBox n - 1 ) 3 RowBox ( RowBox RowBox RowBox - 5 SuperscriptBox n 2 + RowBox 12 n + RowBox 24 RowBox ( RowBox RowBox log ( RowBox SqrtBox n + 1 ) - RowBox log ( RowBox 1 - SqrtBox n ) ) SqrtBox n + RowBox 12 RowBox ( RowBox RowBox RowBox ( RowBox n - 6 ) n - 3 ) RowBox log ( 2 ) + 21 ) SuperscriptBox RowBox ( RowBox m - 1 ) 2 + ) /; RowBox RowBox ( RowBox m 1 ) RowBox ¬ RowBox FractionBox RowBox RowBox 2 RowBox Re ( z ) + π RowBox 4 π TagBox Function TraditionalForm [/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[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[SqrtBox["n"], " ", RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox["n"], " ", RowBox[List["Sin", "[", "z", "]"]]]], "]"]]]], "-", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], RowBox[List["n", "-", "1"]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["ArcTanh", "[", RowBox[List["Sin", "[", "z", "]"]], "]"]]]], "-", RowBox[List["2", " ", SqrtBox["n"], " ", RowBox[List["ArcTanh", "[", RowBox[List[SqrtBox["n"], " ", RowBox[List["Sin", "[", "z", "]"]]]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02