To get a numerical evaluation of the first (K) and second (E) complete elliptic integrals:
$$K(k)=\int_0^1\frac{dt}{(1-t^2)^{1/2}(1-k^2t^2)^{1/2}}, \ \ \ \ \ E(k)=\int_0^1\frac{(1-k^2t^2)^{1/2}}{(1-t^2)^{1/2}}dt$$
in a left neighbourhood of the point $k=1$. What numerical methods do you recommend to get a "good" approximation of K and E in a left neighbourhood of the point $k=1$?