# §29.20(i) Lamé Functions

The eigenvalues $\mathop{a^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, $\mathop{b^{m}_{\nu}\/}\nolimits\!\left(k^{2}\right)$, and the Lamé functions $\mathop{\mathit{Ec}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, $\mathop{\mathit{Es}^{m}_{\nu}\/}\nolimits\!\left(z,k^{2}\right)$, can be calculated by direct numerical methods applied to the differential equation (29.2.1); see §3.7. The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5).

A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i). Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. (Equation (29.6.3) serves as a check.) The Fourier series may be summed using Clenshaw’s algorithm; see §3.11(ii). For further information see Jansen (1977).

A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv). These matrices are the same as those provided in §29.15(i) for the computation of Lamé polynomials with the difference that $n$ has to be chosen sufficiently large. The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as $n\to\infty$. The numerical computations described in Jansen (1977) are based in part upon this method.

A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. See (f) of §28.34(ii).

# §29.20(ii) Lamé Polynomials

The eigenvalues corresponding to Lamé polynomials are computed from eigenvalues of the finite tridiagonal matrices $\mathbf{M}$ given in §29.15(i), using methods described in §3.2(vi) and Ritter (1998). The corresponding eigenvectors yield the coefficients in the finite Fourier series for Lamé polynomials. §29.15(i) includes formulas for normalizing the eigenvectors.

# §29.20(iii) Zeros

Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). Alternatively, the zeros can be found by locating the maximum of function $g$ in (29.12.11).