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 EllipticPi

 http://functions.wolfram.com/08.06.06.0017.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] Csc[z] ((-((Pi Sqrt[(-n) Sin[z]^2])/(2 n))) (1 + (1 + m)/(2 n) + (3 + 2 m + 3 m^2)/(8 n^2) + \[Ellipsis]) + ((1/n) AppellF1[-(1/2), 1/2, 1/2, 1/2, m Sin[z]^2, Sin[z]^2] + (1/(3 n^2 z^2)) AppellF1[-(3/2), 1/2, 1/2, -(1/2), m Sin[z]^2, Sin[z]^2] + (1/(5 n^3 z^4)) AppellF1[-(5/2), 1/2, 1/2, -(3/2), m Sin[z]^2, Sin[z]^2] + \[Ellipsis])) + Round[Re[z]/Pi] ((Pi/Sqrt[-n]) (1 + (1 + m)/(2 n) + (3 + 2 m + 3 m^2)/(8 n^2) + \[Ellipsis]) + (1/(2 n)) (4 (EllipticE[m] - EllipticK[m]) - (-8 (1 + m) EllipticE[m] + 4 (2 + m) EllipticK[m])/(3 n) + (8 ((8 + 7 m + 8 m^2) EllipticE[m] - (8 + 3 m + 4 m^2) EllipticK[m]))/ (15 n^2) + \[Ellipsis])) /; (Abs[n] -> Infinity)

 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["Csc", "[", "z", "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List[RowBox[List["-", "n"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], RowBox[List["2", " ", "n"]]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "+", "m"]], RowBox[List["2", " ", "n"]]], "+", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["8", " ", SuperscriptBox["n", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "n"], RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["3", " ", SuperscriptBox["n", "2"], " ", SuperscriptBox["z", "2"]]]], RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["5", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox["z", "4"]]]], RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["5", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["3", "2"]]], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]]]], "+", "\[Ellipsis]"]], ")"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[RowBox[List[" ", "\[Pi]"]], SqrtBox[RowBox[List["-", "n"]]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "+", "m"]], RowBox[List["2", " ", "n"]]], "+", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["8", " ", SuperscriptBox["n", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", RowBox[List["2", " ", "n"]]], RowBox[List["(", RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], "-", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["2", "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], RowBox[List["3", " ", "n"]]], "+", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["8", "+", RowBox[List["7", " ", "m"]], "+", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["8", "+", RowBox[List["3", " ", "m"]], "+", RowBox[List["4", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], RowBox[List["15", " ", SuperscriptBox["n", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 Π ( n ; z m ) ( - 1 ) Re ( z ) π csc ( z ) ( ( 1 n F 1 AppellF1 ( - 1 2 ; 1 2 , 1 2 ; 1 2 ; m sin 2 ( z ) , sin 2 ( z ) ) + 1 3 n 2 z 2 F 1 AppellF1 ( - 3 2 ; 1 2 , 1 2 ; - 1 2 ; m sin 2 ( z ) , sin 2 ( z ) ) + 1 5 n 3 z 4 F 1 AppellF1 ( - 5 2 ; 1 2 , 1 2 ; - 3 2 ; m sin 2 ( z ) , sin 2 ( z ) ) + ) - π - n sin 2 ( z ) 2 n ( 1 + m + 1 2 n + 3 m 2 + 2 m + 3 8 n 2 + ) ) + Re ( z ) π ( π - n ( 1 + m + 1 2 n + 3 m 2 + 2 m + 3 8 n 2 + ) + 1 2 n ( 4 ( E ( m ) - K ( m ) ) - 4 ( m + 2 ) K ( m ) - 8 ( m + 1 ) E ( m ) 3 n + 8 ( ( 8 m 2 + 7 m + 8 ) E ( m ) - ( 4 m 2 + 3 m + 8 ) K ( m ) ) 15 n 2 + ) ) /; ( "\[LeftBracketingBar]" n "\[RightBracketingBar]" "\[Rule]" ) FormBox RowBox RowBox RowBox Π ( RowBox n ; RowBox z m ) RowBox RowBox SuperscriptBox RowBox ( RowBox - 1 ) ErrorBox RowBox FractionBox RowBox Re ( z ) π RowBox csc ( z ) RowBox ( RowBox RowBox ( RowBox RowBox FractionBox 1 n RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox RowBox - FractionBox 1 2 ; FractionBox 1 2 , RowBox FractionBox 1 2 ; FractionBox 1 2 ; RowBox m RowBox SuperscriptBox sin 2 ( z ) , RowBox SuperscriptBox sin 2 ( z ) ) + RowBox FractionBox 1 RowBox 3 SuperscriptBox n 2 SuperscriptBox z 2 RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox RowBox - FractionBox 3 2 ; FractionBox 1 2 , RowBox FractionBox 1 2 ; RowBox - FractionBox 1 2 ; RowBox m RowBox SuperscriptBox sin 2 ( z ) , RowBox SuperscriptBox sin 2 ( z ) ) + RowBox FractionBox 1 RowBox 5 SuperscriptBox n 3 SuperscriptBox z 4 RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox RowBox - FractionBox 5 2 ; FractionBox 1 2 , RowBox FractionBox 1 2 ; RowBox - FractionBox 3 2 ; RowBox m RowBox SuperscriptBox sin 2 ( z ) , RowBox SuperscriptBox sin 2 ( z ) ) + ) - RowBox FractionBox RowBox π SqrtBox RowBox RowBox - n RowBox SuperscriptBox sin 2 ( z ) RowBox 2 n RowBox ( RowBox 1 + FractionBox RowBox m + 1 RowBox 2 n + FractionBox RowBox RowBox 3 SuperscriptBox m 2 + RowBox 2 m + 3 RowBox 8 SuperscriptBox n 2 + ) ) + RowBox RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox RowBox FractionBox π SqrtBox RowBox - n RowBox ( RowBox 1 + FractionBox RowBox m + 1 RowBox 2 n + FractionBox RowBox RowBox 3 SuperscriptBox m 2 + RowBox 2 m + 3 RowBox 8 SuperscriptBox n 2 + ) + RowBox FractionBox 1 RowBox 2 n RowBox ( RowBox RowBox 4 RowBox ( RowBox RowBox E ( m ) - RowBox K ( m ) ) - FractionBox RowBox RowBox 4 RowBox ( RowBox m + 2 ) RowBox K ( m ) - RowBox 8 RowBox ( RowBox m + 1 ) RowBox E ( m ) RowBox 3 n + FractionBox RowBox 8 RowBox ( RowBox RowBox RowBox ( RowBox RowBox 8 SuperscriptBox m 2 + RowBox 7 m + 8 ) RowBox E ( m ) - RowBox RowBox ( RowBox RowBox 4 SuperscriptBox m 2 + RowBox 3 m + 8 ) RowBox K ( m ) ) RowBox 15 SuperscriptBox n 2 + ) ) /; RowBox ( RowBox RowBox n ) 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["Csc", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List[RowBox[List["-", "n"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "+", "m"]], RowBox[List["2", " ", "n"]]], "+", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["8", " ", SuperscriptBox["n", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["2", " ", "n"]]]]], "+", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], "n"], "+", FractionBox[RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], RowBox[List["3", " ", SuperscriptBox["n", "2"], " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["AppellF1", "[", RowBox[List[RowBox[List["-", FractionBox["5", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", RowBox[List["-", FractionBox["3", "2"]]], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]], "]"]], RowBox[List["5", " ", SuperscriptBox["n", "3"], " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "+", "m"]], RowBox[List["2", " ", "n"]]], "+", FractionBox[RowBox[List["3", "+", RowBox[List["2", " ", "m"]], "+", RowBox[List["3", " ", SuperscriptBox["m", "2"]]]]], RowBox[List["8", " ", SuperscriptBox["n", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], SqrtBox[RowBox[List["-", "n"]]]], "+", FractionBox[RowBox[List[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticE", "[", "m", "]"]], "-", RowBox[List["EllipticK", "[", "m", "]"]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List["4", " ", RowBox[List["(", RowBox[List["2", "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], RowBox[List["3", " ", "n"]]], "+", FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["8", "+", RowBox[List["7", " ", "m"]], "+", RowBox[List["8", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["8", "+", RowBox[List["3", " ", "m"]], "+", RowBox[List["4", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], RowBox[List["15", " ", SuperscriptBox["n", "2"]]]], "+", "\[Ellipsis]"]], RowBox[List["2", " ", "n"]]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02