§36.15 Methods of Computation

Close to the origin $𝐱=\mathrm{𝟎}$ of parameter space, the series in §36.8 can be used.

Far from the bifurcation set, the leading-order asymptotic formulas of §36.11 reproduce accurately the form of the function, including the geometry of the zeros described in §36.7. Close to the bifurcation set but far from $𝐱=\mathrm{𝟎}$, the uniform asymptotic approximations of §36.12 can be used.

Direct numerical evaluation can be carried out along a contour that runs along the segment of the real $t$-axis containing all real critical points of $\mathrm{\Phi }$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\exp \left(\mathrm{i}\mathrm{\Phi }\right)$. (For the umbilics, representations as one-dimensional integrals (§36.2) are used.) For details, see Connor and Curtis (1982), Kirk et al. (2000), Feldbrugge et al. (2023), Gibbs et al. (2024) and Gibbs (2025). There is considerable freedom in the choice of contour deformation as it runs between regions containing saddles and endpoints in valleys at infinity.

This can be carried out by direct numerical evaluation of canonical integrals along a finite segment of the real axis including all real critical points of $\mathrm{\Phi }$, with contributions from the contour outside this range approximated by the first terms of an asymptotic series associated with the endpoints. See Berry et al. (1979).

For numerical solution of partial differential equations satisfied by the canonical integrals see Connor et al. (1983).