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 EllipticPi

 http://functions.wolfram.com/08.06.06.0014.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (Sin[z] AppellF1[1/2, 3/2, 1/2, 3/2, Sin[z]^2, m Sin[z]^2] + (Sin[z]^3/3) AppellF1[3/2, 5/2, 1/2, 5/2, Sin[z]^2, m Sin[z]^2] (n - 1) + (Sin[z]^5/5) AppellF1[5/2, 7/2, 1/2, 7/2, Sin[z]^2, m Sin[z]^2] (n - 1)^2 + \[Ellipsis]) + Round[Re[z]/Pi] (Pi/(Sqrt[1 - m/n] Sqrt[1 - n]) + (2 EllipticE[m])/(m - 1) + 2 EllipticK[m] + ((2 ((1 + m) EllipticE[m] + (-1 + m) EllipticK[m]))/(3 (-1 + m)^2)) (n - 1) - (2/(15 (-1 + m)^3)) ((-3 - 7 m + 2 m^2) EllipticE[m] - (-3 + 2 m + m^2) EllipticK[m]) (n - 1)^2 + \[Ellipsis]) /; (n -> 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[RowBox[List[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "3"], "3"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["5", "2"], ",", FractionBox["1", "2"], ",", FractionBox["5", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"], "5"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox["7", "2"], ",", FractionBox["1", "2"], ",", FractionBox["7", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["m", "n"]]]], SqrtBox[RowBox[List["1", "-", "n"]]], " "]]], "+", FractionBox[RowBox[List["2", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["m", "-", "1"]]], "+", RowBox[List["2", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["2", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"]]]], RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]], "-", RowBox[List[FractionBox["2", RowBox[List["15", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["7", " ", "m"]], "+", RowBox[List["2", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "m"]], "+", SuperscriptBox["m", "2"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[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 ( sin ( z ) F 1 AppellF1 ( 1 2 ; 3 2 , 1 2 ; 3 2 ; sin 2 ( z ) , m sin 2 ( z ) ) + sin 3 ( z ) 3 F 1 AppellF1 ( 3 2 ; 5 2 , 1 2 ; 5 2 ; sin 2 ( z ) , m sin 2 ( z ) ) ( n - 1 ) + sin 5 ( z ) 5 F 1 AppellF1 ( 5 2 ; 7 2 , 1 2 ; 7 2 ; sin 2 ( z ) , m sin 2 ( z ) ) ( n - 1 ) 2 + ) + Re ( z ) π ( π 1 - m n 1 - n + 2 E ( m ) m - 1 + 2 K ( m ) + 2 ( m + 1 ) E ( m ) + ( m - 1 ) K ( m ) 3 ( m - 1 ) 2 ( n - 1 ) - 2 ( 2 m 2 - 7 m - 3 ) E ( m ) - ( m 2 + 2 m - 3 ) K ( m ) 15 ( m - 1 ) 3 ( n - 1 ) 2 + ) /; ( n "\[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 RowBox RowBox sin ( z ) RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox FractionBox 1 2 ; FractionBox 3 2 , RowBox FractionBox 1 2 ; FractionBox 3 2 ; RowBox SuperscriptBox sin 2 ( z ) , RowBox m RowBox SuperscriptBox sin 2 ( z ) ) + RowBox FractionBox RowBox SuperscriptBox sin 3 ( z ) 3 RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox FractionBox 3 2 ; FractionBox 5 2 , RowBox FractionBox 1 2 ; FractionBox 5 2 ; RowBox SuperscriptBox sin 2 ( z ) , RowBox m RowBox SuperscriptBox sin 2 ( z ) ) RowBox ( RowBox n - 1 ) + RowBox FractionBox RowBox SuperscriptBox sin 5 ( z ) 5 RowBox InterpretationBox SubscriptBox F 1 AppellF1 Rule Editable Rule Selectable ( RowBox RowBox FractionBox 5 2 ; FractionBox 7 2 , RowBox FractionBox 1 2 ; FractionBox 7 2 ; RowBox SuperscriptBox sin 2 ( z ) , RowBox m RowBox SuperscriptBox sin 2 ( z ) ) SuperscriptBox RowBox ( RowBox n - 1 ) 2 + ) + RowBox RowBox FractionBox RowBox Re ( z ) π RowBox ( RowBox FractionBox π RowBox SqrtBox RowBox 1 - FractionBox m n SqrtBox RowBox 1 - n + FractionBox RowBox 2 RowBox E ( m ) RowBox m - 1 + RowBox 2 RowBox K ( m ) + RowBox 2 FractionBox RowBox RowBox RowBox ( RowBox m + 1 ) RowBox E ( m ) + RowBox RowBox ( RowBox m - 1 ) RowBox K ( m ) RowBox 3 SuperscriptBox RowBox ( RowBox m - 1 ) 2 RowBox ( RowBox n - 1 ) - RowBox 2 FractionBox RowBox RowBox RowBox ( RowBox RowBox 2 SuperscriptBox m 2 - RowBox 7 m - 3 ) RowBox E ( m ) - RowBox RowBox ( RowBox SuperscriptBox m 2 + RowBox 2 m - 3 ) RowBox K ( m ) RowBox 15 SuperscriptBox RowBox ( RowBox m - 1 ) 3 SuperscriptBox RowBox ( RowBox n - 1 ) 2 + ) /; RowBox RowBox ( RowBox n 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[RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "3"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "3"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["3", "2"], ",", FractionBox["5", "2"], ",", FractionBox["1", "2"], ",", FractionBox["5", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]], "+", RowBox[List[FractionBox["1", "5"], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["5", "2"], ",", FractionBox["7", "2"], ",", FractionBox["1", "2"], ",", FractionBox["7", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["m", "n"]]]], " ", SqrtBox[RowBox[List["1", "-", "n"]]]]]], "+", FractionBox[RowBox[List["2", " ", RowBox[List["EllipticE", "[", "m", "]"]]]], RowBox[List["m", "-", "1"]]], "+", RowBox[List["2", " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "m"]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]]]], RowBox[List["3", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "2"]]]], "-", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["7", " ", "m"]], "+", RowBox[List["2", " ", SuperscriptBox["m", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", "m", "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", RowBox[List["2", " ", "m"]], "+", SuperscriptBox["m", "2"]]], ")"]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "2"]]], RowBox[List["15", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]], "3"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", "1"]], ")"]], "&&", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["2", " ", RowBox[List["Re", "[", "z", "]"]]]]]], RowBox[List["4", " ", "\[Pi]"]]], "\[Element]", "Integers"]]]]]]]]]]]]

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

 2007-05-02