If (29.2.1) admits a Lamé polynomial solution $E$, then a second linearly independent solution $F$ is given by

29.17.1 | $$F(z)=E(z){\int}_{\mathrm{i}{K}^{\prime}}^{z}\frac{du}{{(E(u))}^{2}}.$$ | ||

For properties of these solutions see Arscott (1964b, §9.7), Erdélyi et al. (1955, §15.5.1), Shail (1980), and Sleeman (1966b).