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1: 19.38 Approximations
§19.38 Approximations
2: 18.15 Asymptotic Approximations
§18.15 Asymptotic Approximations
§18.15(i) Jacobi
§18.15(ii) Ultraspherical
§18.15(iii) Legendre
§18.15(iv) Laguerre
3: 19.27 Asymptotic Approximations and Expansions
Although they are obtained (with some exceptions) by approximating uniformly the integrand of each elliptic integral, some occur also as the leading terms of known asymptotic series with error bounds (Wong (1983, §4), Carlson and Gustafson (1985), López (2000, 2001)). …
4: 2.4 Contour Integrals
§2.4(i) Watson’s Lemma
For examples see Olver (1997b, pp. 315–320).
§2.4(iii) Laplace’s Method
§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method
§2.4(vi) Other Coalescing Critical Points
5: 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 ) . …The order of the approximating Bessel functions, or modified Bessel functions, is 1 / ( λ + 2 ) , except in the case when g ( z ) has a double pole at z 0 . … Then for large u asymptotic approximations of the solutions w can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on u and α ). …
6: 10.21 Zeros
The zeros of any cylinder function or its derivative are simple, with the possible exceptions of z = 0 in the case of the functions, and z = 0 , ± ν in the case of the derivatives. … The approximations that follow in §10.21(viii) do not suffer from this drawback.
§10.21(viii) Uniform Asymptotic Approximations for Large Order
Except possibly for x = 0 their zeros are the same as those of J ν ( x ) and Y ν ( x ) , respectively. …This information includes asymptotic approximations analogous to those given in §§10.21(vi), 10.21(vii), and 10.21(x). …
7: 2.7 Differential Equations
§2.7(iii) Liouville–Green (WKBJ) Approximation
For irregular singularities of nonclassifiable rank, a powerful tool for finding the asymptotic behavior of solutions, complete with error bounds, is as follows:
Liouville–Green Approximation Theorem
By approximatingThe first of these references includes extensions to complex variables and reversions for zeros. …
8: 36.13 Kelvin’s Ship-Wave Pattern
Then with g denoting the acceleration due to gravity, the wave height is approximately given by … When ρ > 1 , that is, everywhere except close to the ship, the integrand oscillates rapidly. … 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
9: 2.6 Distributional Methods
§2.6 Distributional Methods
§2.6(ii) Stieltjes Transform
Corresponding results for the generalized Stieltjes transform …An application has been given by López (2000) to derive asymptotic expansions of standard symmetric elliptic integrals, complete with error bounds; see §19.27(vi). … For rigorous derivations of these results and also order estimates for δ n ( x ) , see Wong (1979) and Wong (1989, Chapter 6).
10: 15.19 Methods of Computation
For z it is possible to use the linear transformations in such a way that the new arguments lie within the unit circle, except when z = e ± π i / 3 . … Gauss quadrature approximations are discussed in Gautschi (2002b). … Initial values for moderate values of | a | and | b | can be obtained by the methods of §15.19(i), and for large values of | a | , | b | , or | c | via the asymptotic expansions of §§15.12(ii) and 15.12(iii). …