§29.17 Other Solutions

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

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

Algebraic Lamé functions are solutions of (29.2.1) when $\nu$ is half an odd integer. They are algebraic functions of $\mathrm{sn}(z,k)$, $\mathrm{cn}(z,k)$, and $\mathrm{dn}(z,k)$, and have primitive period $8K$. See Erdélyi (1941c), Ince (1940b), and Lambe (1952).

Lamé–Wangerin functions are solutions of (29.2.1) with the property that ${(\mathrm{sn}(z,k))}^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}{K}^{\prime }$ to $2K+\mathrm{i}{K}^{\prime }$. See Erdélyi et al. (1955, §15.6).