Numerical evaluation of the first (K) and second (E) complete elliptic integrals 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 $k=1$?

$\quad$ Use the trigonometric-integral expressions for these functions, expand them using the binomial series, switch the order of summation and integration, then make use of Wallis' integrals in order to arrive at the well-known but slow-converging infinite series for each, also presented in the article. Use the recurrence relation $T_n=\bigg(\dfrac{2n-1}{2n}k\bigg)^2\cdot a_n\cdot T_{n-1}$, where $a_n=1$ for $K(x)$, and $\dfrac{2n-3}{2n-1}$ for $E(x)$.