# uniform asymptotic approximation

(0.003 seconds)

## 1—10 of 56 matching pages

##### 1: 14.26 Uniform Asymptotic Expansions

###### §14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\mathit{Q}}_{\nu}^{\mu}\left(x\right)$ for $$ are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). … ►See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.##### 2: 28.26 Asymptotic Approximations for Large $q$

##### 3: 36.15 Methods of Computation

…
►Close to the bifurcation set but far from $\mathbf{x}=\mathbf{0}$, the uniform asymptotic approximations of §36.12 can be used.
…

##### 4: 28.8 Asymptotic Expansions for Large $q$

###### §28.8 Asymptotic Expansions for Large $q$

… ►###### Barrett’s Expansions

… ►###### Dunster’s Approximations

►Dunster (1994a) supplies uniform asymptotic approximations for numerically satisfactory pairs of solutions of Mathieu’s equation (28.2.1). … ►##### 5: 14.15 Uniform Asymptotic Approximations

###### §14.15 Uniform Asymptotic Approximations

►###### §14.15(i) Large $\mu $, Fixed $\nu $

… ►For asymptotic expansions and explicit error bounds, see Dunster (2003b). ►###### §14.15(iii) Large $\nu $, Fixed $\mu $

… ►##### 6: 10.72 Mathematical Applications

…
►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
…
►If $f(z)$ has a double zero ${z}_{0}$, or more generally ${z}_{0}$ is a zero of order $m$, $m=2,3,4,\mathrm{\dots}$, then uniform asymptotic approximations (but

*not*expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order $1/(m+2)$. …##### 7: 36.13 Kelvin’s Ship-Wave Pattern

…
►The disturbance $z(\rho ,\varphi )$ can be approximated by the method of uniform asymptotic approximation for the case of two coalescing stationary points (36.12.11), using the fact that ${\theta}_{\pm}(\varphi )$ are real for $$ and complex for $|\varphi |>{\varphi}_{c}$.
…
►
…

##### 8: 7.20 Mathematical Applications

…
►For applications of the complementary error function in uniform asymptotic approximations of integrals—saddle point coalescing with a pole or saddle point coalescing with an endpoint—see Wong (1989, Chapter 7), Olver (1997b, Chapter 9), and van der Waerden (1951).
…

##### 9: Bibliography R

…
►
Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow.
Studies in Appl. Math. 53, pp. 91–110.
►
Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory.
Studies in Appl. Math. 53, pp. 217–224.
…

##### 10: 9.16 Physical Applications

…
►This reference provides several examples of applications to problems in quantum mechanics in which Airy functions give uniform asymptotic approximations, valid in the neighborhood of a turning point.
…