exponentially-small contributions

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1: 10.69 Uniform Asymptotic Expansions for Large Order

… ►Accuracy in (10.69.2) and (10.69.4) can be increased by including exponentially-small contributions as in (10.67.3), (10.67.4), (10.67.7), and (10.67.8) with $x$ replaced by $\nu x$.

2: 10.67 Asymptotic Expansions for Large Argument

… ►The contributions of the terms in ${\ker }_{\nu }x$, ${\mathrm{kei}}_{\nu }x$, ${\ker }_{\nu }^{\prime }x$, and ${\mathrm{kei}}_{\nu }^{\prime }x$ on the right-hand sides of (10.67.3), (10.67.4), (10.67.7), and (10.67.8) are exponentially small compared with the other terms, and hence can be neglected in the sense of Poincaré asymptotic expansions (§2.1(iii)). … …

3: 2.11 Remainder Terms; Stokes Phenomenon

… ►In effect, (2.11.7) “corrects” (2.11.6) by introducing a term that is relatively exponentially small in the neighborhood of $\mathrm{ph}z=\pi$, is increasingly significant as $\mathrm{ph}z$ passes from $\pi$ to $\frac{3}{2}\pi$, and becomes the dominant contribution after $\mathrm{ph}z$ passes $\frac{3}{2}\pi$. … ►Hence from §7.12(i) $\mathrm{erfc}\left(\sqrt{\frac{1}{2}\rho }c(\theta )\right)$ is of the same exponentially-small order of magnitude as the contribution from the other terms in (2.11.15) when $\rho$ is large. … ►In particular, on the ray $\theta =\pi$ greatest accuracy is achieved by (a) taking the average of the expansions (2.11.6) and (2.11.7), followed by (b) taking account of the exponentially-small contributions arising from the terms involving $h_{2s}(\theta ,\alpha )$ in (2.11.15). … ►As these lines are crossed exponentially-small contributions, such as that in (2.11.7), are “switched on” smoothly, in the manner of the graph in Figure 2.11.1. …

4: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . …

5: 8.20 Asymptotic Expansions of $E_p\left(z\right)$

… ►Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …

6: 36.11 Leading-Order Asymptotics

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7: 3.5 Quadrature

… ►In particular, when $k=\mathrm{\infty }$ the error term is an exponentially-small function of $1/h$, and in these circumstances the composite trapezoidal rule is exceptionally efficient. … ►A frequent problem with contour integrals is heavy cancellation, which occurs especially when the value of the integral is exponentially small compared with the maximum absolute value of the integrand. To avoid cancellation we try to deform the path to pass through a saddle point in such a way that the maximum contribution of the integrand is derived from the neighborhood of the saddle point. … ►When $\lambda$ is large the integral becomes exponentially small, and application of the quadrature rule of §3.5(viii) is useless. … ►In more advanced methods points are sampled from a probability distribution, so that they are concentrated in regions that make the largest contribution to the integral. …