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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 ν - μ ( x ) and Q ν μ ( x ) for 1 < x < 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 x = 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 μ , Fixed ν
For asymptotic expansions and explicit error bounds, see Dunster (2003b).
§14.15(iii) Large ν , Fixed μ
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 , , 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 ( ρ , ϕ ) 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 θ ± ( ϕ ) are real for | ϕ | < ϕ c and complex for | ϕ | > ϕ 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 = ρ cos ϕ , y = ρ sin ϕ . 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: Bibliography R
  • W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.
  • W. H. Reid (1974b) 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. …