ELLIPTIC FUNCTIONS

    In an earlier post I have discussed the elliptic exponential function                                    E(iu, k) = cn(u, k) +i sn(u, k)      By replacing u by -iu in this function we obtain the function E(u, k). When q approaches 0,  E(iu, k)  reduces to the exponential function e^iu = cos u + i sin u and E(u, k) reduces to e^u.  As is well-known, the inverse of the exponential function v = e^u is the logarithm function ln v =u, such that ln(1+v) = Σ (-1)^{n-1}. v^n /n                                                                                                       (1...

      Contd.   3.      4.((θ_2)^5) ((θ_3)^4) ((θ_4)^2) + (θ_2) ((θ_3)^8) ((θ_4)^2) + 6.(θ_2^(2)) (θ_4) (θ_4^(2))                                                            - 6.(θ_2)((θ_4^(2))^2) + (θ_2) (θ_4) (θ_4^(4))                                                               = (θ_2^(4)) ((θ_4)^2)   4.      5. (θ_1^(1)) ( θ_4) ( θ_4^(4)) + 10.( θ_1^(3))( θ_4^(2) ) ( θ_4) - 30. ( θ_1^(1)) ( θ_4^(2) )^2            + ( θ_2)(( θ_3)^9) (( θ_4)^3) + 14. (( θ_2)^5) (( θ_3)^5) (( θ_4)^3) + (( θ_2)^9) ( θ_3) (( θ_4)^3)                      ...

  Let  θ_1(z, q),  θ_2(z, q),  θ_3(z, q)  and θ_4(z, q) denote, as usual, the four basic Jacobian theta functions.      Let θ_2, θ_3, and θ_4 denote θ_2(0, q), θ_3(0, q) and θ_4(0, q); and let θ_1^(n), θ_2^(n), θ_3^(n) and θ_4^(n) denote, respectively, the values of the n-th derivatives of the four functions at z=0 .  Note that (θ_2)^m, (θ_3)^m , and (θ_4)^m denote the ordinary m-th powers. ( Bracketed exponents denote orders of derivatives and un-bracketed exponents denote powers, as usual.)   The following identity of Jacobi is well-known.                                                       (θ_2) (θ_3) (θ_4) = θ_1^(1)      This belongs to an infinite class of identities involving the four theta functions and their n-th derivatives for various values of n.   ...