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 EllipticPi

 http://functions.wolfram.com/08.06.13.0003.01

 Input Form

 Derivative[1][w][z]^6 (-1 + Derivative[1][w][z]^2) (27 m^2 n + 4 (m - n)^3 Derivative[1][w][z]^2)^2 (1 + (-1 + m) (-1 + n)^2 Derivative[1][w][z]^2) + Derivative[1][w][z]^4 (-243 m^4 n^2 - 9 m^2 n (-3 m (-12 + n) n^2 - 10 n^3 + 3 m^2 n (4 + n (-16 + 3 n)) + m^3 (16 + 6 n (-5 + 3 n))) Derivative[1][w][z]^2 - 2 (m - n)^2 (2 m (-10 + n) n^3 + 4 n^4 - 3 m^2 (-9 + n) n^2 (-1 + 2 n) + 4 m^4 (-1 + n) (-2 + 3 n) + m^3 n (8 + 3 (5 - 11 n) n)) Derivative[1][w][z]^4 + 8 (-1 + m) (m - n)^5 (-1 + n) n Derivative[1][w][z]^6) Derivative[2][w][z]^2 + n Derivative[1][w][z]^2 (-27 m^4 n - m^2 (m - n)^2 (-10 n + m (-8 + 9 n)) Derivative[1][w][z]^2 + (-1 + m) (m - n)^4 n Derivative[1][w][z]^4) Derivative[2][w][z]^4 - m^4 n^2 Derivative[2][w][z]^6 == 0 /; w[z] == EllipticPi[n, z, m]

 Standard Form

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 MathML Form

 ( w ( z ) 2 - 1 ) ( 4 w ( z ) 2 ( m - n ) 3 + 27 m 2 n ) 2 ( ( m - 1 ) ( n - 1 ) 2 w ( z ) 2 + 1 ) w ( z ) 6 + ( 8 ( m - 1 ) ( m - n ) 5 ( n - 1 ) n w ( z ) 6 - 2 ( m - n ) 2 ( 4 ( n - 1 ) ( 3 n - 2 ) m 4 + n ( 3 n ( 5 - 11 n ) + 8 ) m 3 - 3 ( n - 9 ) n 2 ( 2 n - 1 ) m 2 + 2 ( n - 10 ) n 3 m + 4 n 4 ) w ( z ) 4 - 9 m 2 n ( ( 6 n ( 3 n - 5 ) + 16 ) m 3 + 3 n ( n ( 3 n - 16 ) + 4 ) m 2 - 3 ( n - 12 ) n 2 m - 10 n 3 ) w ( z ) 2 - 243 m 4 n 2 ) w ′′ ( z ) 2 w ( z ) 4 + n ( - 27 n m 4 - ( m - n ) 2 ( m ( 9 n - 8 ) - 10 n ) w ( z ) 2 m 2 + ( m - 1 ) ( m - n ) 4 n w ( z ) 4 ) w ′′ ( z ) 4 w ( z ) 2 - m 4 n 2 w ′′ ( z ) 6 0 /; w ( z ) Π ( n ; z m ) Condition z w z 2 -1 4 z w z 2 m -1 n 3 27 m 2 n 2 m -1 n -1 2 z w z 2 1 z w z 6 8 m -1 m -1 n 5 n -1 n z w z 6 -1 2 m -1 n 2 4 n -1 3 n -2 m 4 n 3 n 5 -1 11 n 8 m 3 -1 3 n -9 n 2 2 n -1 m 2 2 n -10 n 3 m 4 n 4 z w z 4 -1 9 m 2 n 6 n 3 n -5 16 m 3 3 n n 3 n -16 4 m 2 -1 3 n -12 n 2 m -1 10 n 3 z w z 2 -1 243 m 4 n 2 z 2 w z 2 z w z 4 n -27 n m 4 -1 m -1 n 2 m 9 n -8 -1 10 n z w z 2 m 2 m -1 m -1 n 4 n z w z 4 z 2 w z 4 z w z 2 -1 m 4 n 2 z 2 w z 6 0 w z EllipticPi n z m [/itex]

 Rule Form

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

 2001-10-29