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 EllipticPi

 http://functions.wolfram.com/08.06.06.0073.01

 Input Form

 EllipticPi[n, z, m] \[Proportional] (-(1/Sqrt[m])) EllipticPi[n/m, 1/m] - 2 u EllipticPi[n, m] + ((m Sqrt[2])/((n - m) Sqrt[-1 + m])) Sqrt[Sqrt[-1 + m] (z - Subscript[z, 0])] (1 + ((m^2 + 10 n - m (2 + 9 n))/(12 Sqrt[-1 + m] (m - n))) (z - Subscript[z, 0]) + ((m^2 (4 + m (-4 + 9 m)) + 2 m (44 + m (-68 + 15 m)) n + (292 + m (-628 + 345 m)) n^2)/(480 (-1 + m) (m - n)^2)) (z - Subscript[z, 0])^2 + ((15 m^6 + 920 n^3 - m^5 (26 + 21 n) - 4 m n^2 (-502 + 941 n) + m^4 (-12 + 214 n - 1155 n^2) + 2 m^2 n (68 - 2578 n + 2385 n^2) + m^3 (8 - 284 n + 4258 n^2 - 1911 n^3))/(2688 (-1 + m)^(3/2) (m - n)^3)) (z - Subscript[z, 0])^3 + \[Ellipsis]) /; (z -> Subscript[z, 0]) && Subscript[z, 0] == -ArcCsc[Sqrt[m]] + Pi u && Element[u, Integers]

 Standard Form

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 MathML Form

 Π ( n ; z m ) 1 m Π ( n m 1 m ) + 2 u Π ( n m ) - m 2 - m - 1 ( z - z 0 ) ( n - m ) m - 1 ( 1 - ( m 2 - ( 9 n + 2 ) m + 10 n ) ( z - z 0 ) 12 m - 1 ( m - n ) + ( ( m ( 9 m - 4 ) + 4 ) m 2 + 2 ( m ( 15 m - 68 ) + 44 ) n m + ( m ( 345 m - 628 ) + 292 ) n 2 ) / ( 480 ( m - 1 ) ( m - n ) 2 ) ( z - z 0 ) 2 - ( 15 m 6 - ( 21 n + 26 ) m 5 + ( - 1155 n 2 + 214 n - 12 ) m 4 + ( - 1911 n 3 + 4258 n 2 - 284 n + 8 ) m 3 + 2 n ( 2385 n 2 - 2578 n + 68 ) m 2 - 4 n 2 ( 941 n - 502 ) m + 920 n 3 ) / ( 2688 ( m - 1 ) 3 / 2 ( m - n ) 3 ) ( z - z 0 ) 3 + ) /; ( z "\[Rule]" z 0 ) z 0 csc - 1 ( m ) + π u u TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] Condition Proportional EllipticPi n z m 1 m 1 2 -1 EllipticPi n m -1 1 m -1 2 u EllipticPi n m -1 m 2 1 2 -1 m -1 1 2 z -1 Subscript z 0 1 2 n -1 m m -1 1 2 -1 1 -1 m 2 -1 9 n 2 m 10 n z -1 Subscript z 0 12 m -1 1 2 m -1 n -1 m 9 m -4 4 m 2 2 m 15 m -68 44 n m m 345 m -628 292 n 2 480 m -1 m -1 n 2 -1 z -1 Subscript z 0 2 -1 15 m 6 -1 21 n 26 m 5 -1155 n 2 214 n -12 m 4 -1911 n 3 4258 n 2 -1 284 n 8 m 3 2 n 2385 n 2 -1 2578 n 68 m 2 -1 4 n 2 941 n -502 m 920 n 3 2688 m -1 3 2 m -1 n 3 -1 z -1 Subscript z 0 3 Rule z Subscript z 0 Subscript z 0 m 1 2 u u [/itex]

 Rule Form

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

 2007-05-02