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 EllipticPi

 http://functions.wolfram.com/08.06.06.0058.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]] EllipticPi[n, x, m] - ((1/Sqrt[m]) EllipticPi[n/m, 1/m] + (Pi I Sqrt[n])/(2 Sqrt[n - 1] Sqrt[n - m])) (1 - Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]]) + Exp[(-Pi) I Floor[Arg[z - x]/(2 Pi)]] Sum[(1/p!) Sum[(1/j!) Sum[Binomial[j, Subscript[k, 1]] Sum[((-1)^Subscript[k, 1] 2^(Subscript[k, 1] - j) Sin[x]^Subscript[k, 1] (2 Subscript[k, 2] + Subscript[k, 1] - j)^ p Binomial[j - Subscript[k, 1], Subscript[k, 2]] Sum[(Pochhammer[1 - j, 2 (j - i) - 2]/((j - i - 1)! (2 Sin[x])^ (j - 2 i - 1))) Sum[(-1)^Subscript[i, 1] KroneckerDelta[ Subscript[i, 1] + Subscript[i, 2] + Subscript[i, 3] - i] Multinomial[Subscript[i, 1], Subscript[i, 2], Subscript[i, 3]] n^Subscript[i, 1] m^Subscript[i, 3] Pochhammer[ -Subscript[i, 1], Subscript[i, 1]] Pochhammer[1/2, Subscript[i, 2]] Pochhammer[1/2, Subscript[i, 3]] (1 - n Sin[x]^2)^(-1 - Subscript[i, 1]) Cos[x]^ (-2 Subscript[i, 2] - 1) (1 - m Sin[x]^2)^(-(1/2) - Subscript[i, 3]), {Subscript[i, 1], 0, i}, {Subscript[i, 2], 0, i}, {Subscript[i, 3], 0, i}], {i, 0, j - 1}])/E^((1/2) I ((p - 2 Subscript[k, 2] - Subscript[k, 1] + j) Pi + 2 (2 Subscript[k, 2] + Subscript[k, 1] - j) x)), {Subscript[k, 2], 0, j - Subscript[k, 1]}], {Subscript[k, 1], 0, j - 1}], {j, 1, p}] (z - x)^p, {p, 1, Infinity}] /; Element[x, Reals] && Element[m, Reals] && Element[n, Reals] && 1 < m < n && Pi/2 + Pi u < x < Pi (u + 1) - ArcCsc[Sqrt[m]] && Element[u, Integers]

 Standard Form

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"\[Element]", "Integers"]]]]]]]]

 MathML Form

 Π ( n ; z m ) - ( 1 m Π ( n m 1 m ) + π n 2 n - 1 n - m ) ( 1 - - π arg ( z - x ) 2 π ) + Π ( n ; x m ) - π arg ( z - x ) 2 π + - π arg ( z - x ) 2 π p = 1 1 p ! ( j = 1 p 1 j ! k 1 = 0 j - 1 ( j k 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "1"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] k 2 = 0 j - k 1 ( - 1 ) k 1 2 k 1 - j sin k 1 ( x ) ( k 1 + 2 k 2 - j ) p - 1 2 ( π ( j + p - k 1 - 2 k 2 ) + 2 ( k 1 + 2 k 2 - j ) x ) ( j - k 1 k 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "-", SubscriptBox["k", "1"]]], Identity, Rule[Editable, True]]], List[TagBox[SubscriptBox["k", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] i = 0 j - 1 ( 1 - j ) 2 ( j - i ) - 2 ( j - i - 1 ) ! ( 2 sin ( x ) ) j - 2 i - 1 TagBox[FractionBox[SubscriptBox[RowBox[List["(", RowBox[List["1", "-", "j"]], ")"]], RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["j", "-", "i"]], ")"]]]], "-", "2"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["j", "-", "i", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", RowBox[List["sin", "(", "x", ")"]]]], ")"]], RowBox[List["j", "-", RowBox[List["2", "i"]], "-", "1"]]]]]], Pochhammer] i 1 = 0 i i 2 = 0 i i 3 = 0 i ( - 1 ) i 1 δ KroneckerDelta i 1 + i 2 + i 3 - i ( i 1 + i 2 + i 3 ; i 1 , i 2 , i 3 ) TagBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["i", "1"], "+", SubscriptBox["i", "2"], "+", SubscriptBox["i", "3"]]], ";", SubscriptBox["i", "1"]]], ",", SubscriptBox["i", "2"], ",", SubscriptBox["i", "3"]]], ")"]], Multinomial, Rule[Editable, False]] n i 1 m i 3 ( - i 1 ) i 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", SubscriptBox["i", "1"]]], ")"]], SubscriptBox["i", "1"]], Pochhammer] ( 1 2 ) i 2 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], SubscriptBox["i", "2"]], Pochhammer] ( 1 2 ) i 3 TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], SubscriptBox["i", "3"]], Pochhammer] ( 1 - n sin 2 ( x ) ) - i 1 - 1 cos - 2 i 2 - 1 ( x ) ( 1 - m sin 2 ( x ) ) - i 3 - 1 2 ( z - x ) p ) /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] m TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] n TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] 1 < m < n π u + π 2 < x < π ( u + 1 ) - csc - 1 ( m ) u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional EllipticPi n z m -1 1 m 1 2 -1 EllipticPi n m -1 1 m -1 n 1 2 2 n -1 1 2 n -1 m 1 2 -1 1 -1 -1 z -1 x 2 -1 EllipticPi n x m -1 z -1 x 2 -1 -1 z -1 x 2 -1 p 1 1 p -1 j 1 p 1 j -1 Subscript k 1 0 j -1 Binomial j Subscript k 1 Subscript k 2 0 j -1 Subscript k 1 -1 Subscript k 1 2 Subscript k 1 -1 j x Subscript k 1 Subscript k 1 2 Subscript k 2 -1 j p -1 1 2 j p -1 Subscript k 1 -1 2 Subscript k 2 2 Subscript k 1 2 Subscript k 2 -1 j x Binomial j -1 Subscript k 1 Subscript k 2 i 0 j -1 Pochhammer Subscript 1 -1 j 2 j -1 i -2 j -1 i -1 2 x j -1 2 i -1 -1 Subscript i 3 0 i Subscript i 2 0 i Subscript i 1 0 i -1 Subscript i 1 KroneckerDelta Subscript i 1 Subscript i 2 Subscript i 3 -1 i Multinomial Subscript i 1 Subscript i 2 Subscript i 3 n Subscript i 1 m Subscript i 3 Pochhammer -1 Subscript i 1 Subscript i 1 Pochhammer 1 2 Subscript i 2 Pochhammer 1 2 Subscript i 3 1 -1 n x 2 -1 Subscript i 1 -1 x -2 Subscript i 2 -1 1 -1 m x 2 -1 Subscript i 3 -1 1 2 z -1 x p x m n 1 m n u 2 -1 x u 1 -1 m 1 2 u [/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["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[Pi]"]], " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["EllipticPi", "[", RowBox[List["n", ",", "x", ",", "m"]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["EllipticPi", "[", RowBox[List[FractionBox["n", "m"], ",", FractionBox["1", "m"]]], "]"]], SqrtBox["m"]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", SqrtBox["n"]]], RowBox[List["2", " ", SqrtBox[RowBox[List["n", "-", "1"]]], " ", SqrtBox[RowBox[List["n", "-", "m"]]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["\[ExponentialE]", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02