uniform asymptotic approximations

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1: 14.26 Uniform Asymptotic Expansions

§14.26 Uniform Asymptotic Expansions

►The uniform asymptotic approximations given in §14.15 for

P^{-\mu}_{\nu}\left(x\right)

and

\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)

for

1<x<\infty

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 $q$

§28.26 Asymptotic Approximations for Large $q$

… ►

§28.26(ii) Uniform Approximations

…

3: 36.15 Methods of Computation

… ►Close to the bifurcation set but far from

\mathbf{x}=\boldsymbol{{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,\dotsc

, 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,\phi)

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}(\phi)

are real for

|\phi|<\phi_{c}

and complex for

|\phi|>\phi_{c}

. … ►

See accompanying text — Figure 36.13.1: Kelvin’s ship wave pattern, computed from the uniform asymptotic approximation (36.13.8), as a function of $x=\rho\cos\phi$ , $y=\rho\sin\phi$ . Magnify

…

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: 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. …

10: 8.22 Mathematical Applications

… ►For further information on

\zeta_{x}(s)

, including zeros and uniform asymptotic approximations, see Kölbig (1970, 1972a) and Dunster (2006). …