The complete elliptic integrals K(m) and E(m) have very similar infinite series in powers of m, but convergence relies on m being less than say 0.5.
For m near 1, they are expanded in powers of 1-m instead as that is now less than 0.5.
K(m) = K1(1-m) -K2(1-m)Ln(1-m) where K1 and K2 are polynomials I located in Cody 1963, "Chebyshev approximations....." The series for E(m) also with the Ln term (in my question) is mentioned in Cody but he doesn't disclose the polynomial coefficients. I have searched high and low for them and come up blank. Can anybody help? BTW Wolfram doesn't give enough terms to be able to deduce the coefficient sequence.