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 EllipticPi

 http://functions.wolfram.com/08.06.06.0079.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] (-1)^Round[Re[z]/Pi] (((I Sqrt[-Sin[z]^2])/Sin[z]) (EllipticPi[n, m] + (I/Sqrt[-m]) (EllipticK[1/m] - EllipticPi[1/n, 1/m] + I (1 - Sqrt[m/(m - 1)] Sqrt[(m - 1)/m]) (EllipticK[1 - 1/m] + (n/(m - n)) EllipticPi[(m - 1)/(m - n), (m - 1)/m]))) + ((Sqrt[-Sin[z]^2] Sqrt[(-m) Sin[z]^2])/ (n m Sin[z]^5)) Sum[(m^(-k + u) n^(i - u) Pochhammer[1/2, i] Pochhammer[1/2, k - u])/(i! (k - u)! (3 + 2 k))/Sin[z]^(2 k), {k, 0, Infinity}, {u, 0, k}, {i, 0, u}]) + 2 Round[Re[z]/Pi] EllipticPi[n, m] /; (Abs[z] -> Infinity) && ((0 < Arg[n] < Pi/2 && 0 < Arg[m] < Pi) || (Pi/2 < Arg[n] < Pi && 0 < Arg[m] < Pi/2 && (Abs[m] < 1 || (Abs[n] > 1 && Abs[n] > Abs[m]))) || (Pi/2 < Arg[n] < Pi && Pi/2 < Arg[m] < Pi && Abs[n] > Abs[m]))

 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[FractionBox[RowBox[List["\[ImaginaryI]", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], RowBox[List["Sin", "[", "z", "]"]]], RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "+", RowBox[List[FractionBox["\[ImaginaryI]", SqrtBox[RowBox[List["-", "m"]]]], RowBox[List["(", RowBox[List[RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]], "-", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["1", "n"], ",", FractionBox["1", "m"]]], "]"]], "+", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["m", RowBox[List["m", "-", "1"]]]], SqrtBox[FractionBox[RowBox[List["m", "-", "1"]], "m"]]]]]], ")"]], RowBox[List["(", " ", RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]], "+", RowBox[List[FractionBox[RowBox[List["n", " "]], RowBox[List["m", "-", "n"]]], RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], RowBox[List["m", "-", "n"]]], ",", FractionBox[RowBox[List["m", "-", "1"]], "m"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], RowBox[List["n", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "u"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["m", RowBox[List[RowBox[List["-", "k"]], "+", "u"]]], " ", SuperscriptBox["n", RowBox[List["i", "-", "u"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["k", "-", "u"]]]], "]"]]]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "u"]], ")"]], "!"]], RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "k"]]]], ")"]]]]], SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]]]]]]]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "n", "]"]], "<", FractionBox["\[Pi]", "2"]]], "\[And]", RowBox[List["0", "<", RowBox[List["Arg", "[", "m", "]"]], "<", "\[Pi]"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "n", "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List["0", "<", RowBox[List["Arg", "[", "m", "]"]], "<", FractionBox["\[Pi]", "2"]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "<", "1"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ">", "1"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ">", RowBox[List["Abs", "[", "m", "]"]]]]]], ")"]]]], ")"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "n", "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "m", "]"]], "<", "\[Pi]"]], "\[And]", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ">", RowBox[List["Abs", "[", "m", "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 Π ( n ; sin - 1 ( z ) m ) - z 2 z ( - m ( K ( 1 m ) + ( 1 - m m - 1 m - 1 m ) ( K ( 1 - 1 m ) + n m - n Π ( m - 1 m - n m - 1 m ) ) - Π ( 1 n 1 m ) ) ) + Π ( n m ) + - z 2 - m z 2 n m z 5 k = 0 u = 0 k i = 0 u m u - k n i - u ( 1 2 ) i TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "i"], Pochhammer] ( 1 2 ) k - u TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], RowBox[List["k", "-", "u"]]], Pochhammer] i ! ( k - u ) ! ( 2 k + 3 ) z - 2 k /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) ( ( 0 < arg ( n ) < π 2 0 < arg ( m ) < π ) ( π 2 < arg ( n ) < π 0 < arg ( m ) < π 2 ( "\[LeftBracketingBar]" m "\[RightBracketingBar]" < 1 ( "\[LeftBracketingBar]" n "\[RightBracketingBar]" > 1 "\[LeftBracketingBar]" n "\[RightBracketingBar]" > "\[LeftBracketingBar]" m "\[RightBracketingBar]" ) ) ) ( π 2 < arg ( n ) < π π 2 < arg ( m ) < π "\[LeftBracketingBar]" n "\[RightBracketingBar]" > "\[LeftBracketingBar]" m "\[RightBracketingBar]" ) ) Condition Proportional EllipticPi n z m -1 z 2 1 2 z -1 -1 m 1 2 -1 EllipticK 1 m -1 1 -1 m m -1 -1 1 2 m -1 m -1 1 2 EllipticK 1 -1 1 m -1 n m -1 n -1 EllipticPi m -1 m -1 n -1 m -1 m -1 -1 EllipticPi 1 n -1 1 m -1 EllipticPi n m -1 z 2 1 2 -1 m z 2 1 2 n m z 5 -1 i 0 u u 0 k k 0 m u -1 k n i -1 u Pochhammer 1 2 i Pochhammer 1 2 k -1 u i k -1 u 2 k 3 -1 z -2 k Rule z 0 n 2 -1 0 m 2 -1 n 0 m 2 -1 m 1 n 1 n m 2 -1 n 2 -1 m n m [/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["(", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticK", "[", FractionBox["1", "m"], "]"]], "-", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["1", "n"], ",", FractionBox["1", "m"]]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[SqrtBox[FractionBox["m", RowBox[List["m", "-", "1"]]]], " ", SqrtBox[FractionBox[RowBox[List["m", "-", "1"]], "m"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["EllipticK", "[", RowBox[List["1", "-", FractionBox["1", "m"]]], "]"]], "+", FractionBox[RowBox[List["n", " ", RowBox[List["EllipticPi", "[", RowBox[List[FractionBox[RowBox[List["m", "-", "1"]], RowBox[List["m", "-", "n"]]], ",", FractionBox[RowBox[List["m", "-", "1"]], "m"]]], "]"]]]], RowBox[List["m", "-", "n"]]]]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List["-", "m"]]]]]], ")"]]]], RowBox[List["Sin", "[", "z", "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "m"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["u", "=", "0"]], "k"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "u"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", RowBox[List[RowBox[List["-", "k"]], "+", "u"]]], " ", SuperscriptBox["n", RowBox[List["i", "-", "u"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "i"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["k", "-", "u"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List[RowBox[List["-", "2"]], " ", "k"]]]]], RowBox[List[RowBox[List["i", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "u"]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "k"]]]], ")"]]]]]]]]]]]]], RowBox[List["n", " ", "m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "5"]]]]]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["Round", "[", FractionBox[RowBox[List["Re", "[", "z", "]"]], "\[Pi]"], "]"]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "m"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["0", "<", RowBox[List["Arg", "[", "n", "]"]], "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["0", "<", RowBox[List["Arg", "[", "m", "]"]], "<", "\[Pi]"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "n", "]"]], "<", "\[Pi]"]], "&&", RowBox[List["0", "<", RowBox[List["Arg", "[", "m", "]"]], "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "m", "]"]], "<", "1"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ">", "1"]], "&&", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ">", RowBox[List["Abs", "[", "m", "]"]]]]]], ")"]]]], ")"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "n", "]"]], "<", "\[Pi]"]], "&&", RowBox[List[FractionBox["\[Pi]", "2"], "<", RowBox[List["Arg", "[", "m", "]"]], "<", "\[Pi]"]], "&&", RowBox[List[RowBox[List["Abs", "[", "n", "]"]], ">", RowBox[List["Abs", "[", "m", "]"]]]]]], ")"]]]], ")"]]]]]]]]]]

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

 2007-05-02