# §29.17 Other Solutions

## §29.17(i) Second Solution

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

 29.17.1 $F(z)=E(z)\int_{\mathrm{i}{K^{\prime}}}^{z}\frac{\mathrm{d}u}{(E(u))^{2}}.$

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

## §29.17(ii) Algebraic Lamé Functions

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

## §29.17(iii) Lamé–Wangerin Functions

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