# §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 $F(z)=E(z)\int_{\mathrm{i}\!\mathop{{K^{\prime}}\/}\nolimits\!}^{z}\frac{% \mathrm{d}u}{(E(u))^{2}}.$

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 $\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right)$, $\mathop{\mathrm{cn}\/}\nolimits\left(z,k\right)$, and $\mathop{\mathrm{dn}\/}\nolimits\left(z,k\right)$, and have primitive period $8\!\mathop{K\/}\nolimits\!$. 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 $(\mathop{\mathrm{sn}\/}\nolimits\left(z,k\right))^{1/2}w(z)$ is bounded on the line segment from $\mathrm{i}\!\mathop{{K^{\prime}}\/}\nolimits\!$ to $2\!\mathop{K\/}\nolimits\!+\mathrm{i}\!\mathop{{K^{\prime}}\/}\nolimits\!$. See Erdélyi et al. (1955, §15.6).