# §36.15 Methods of Computation

## §36.15(i) Convergent Series

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

## §36.15(ii) Asymptotics

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 $\mathbf{x}=0$, the uniform asymptotic approximations of §36.12 can be used.

## §36.15(iii) Integration along Deformed Contour

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 $\Phi$ and is deformed outside this range so as to reach infinity along the asymptotic valleys of $\mathop{\exp\/}\nolimits\!\left(i\Phi\right)$. (For the umbilics, representations as one-dimensional integrals (§36.2) are used.) For details, see Connor and Curtis (1982) and Kirk et al. (2000). There is considerable freedom in the choice of deformations.

## §36.15(iv) Integration along Finite Contour

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 $\Phi$, 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).

## §36.15(v) Differential Equations

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