asymptotic approximation

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21: 29.16 Asymptotic Expansions

… ►Hargrave and Sleeman (1977) give asymptotic approximations for Lamé polynomials and their eigenvalues, including error bounds. …

22: 35.10 Methods of Computation

… ►For large $\Vert 𝐓\Vert$ the asymptotic approximations referred to in §35.7(iv) are available. …

23: 36.12 Uniform Approximation of Integrals

… ►The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. … ►The leading-order uniform asymptotic approximation is given by … ►For further information concerning integrals with several coalescing saddle points see Arnol’d et al. (1988), Berry and Howls (1993, 1994), Bleistein (1967), Duistermaat (1974), Ludwig (1966), Olde Daalhuis (2000), and Ursell (1972, 1980).

24: 7.20 Mathematical Applications

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§7.20(i) Asymptotics

►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). …

25: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

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§2.4(v) Coalescing Saddle Points: Chester, Friedman, and Ursell’s Method

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§2.4(vi) Other Coalescing Critical Points

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26: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

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§2.11(ii) Connection Formulas

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§2.11(iii) Exponentially-Improved Expansions

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§2.11(vi) Direct Numerical Transformations

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27: 19.12 Asymptotic Approximations

§19.12 Asymptotic Approximations

… ►For the asymptotic behavior of $F(\varphi ,k)$ and $E(\varphi ,k)$ as $\varphi \to \frac{1}{2}\pi -$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007). … ►Asymptotic approximations for $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$, with different variables, are given in Karp et al. (2007). …

28: 2.10 Sums and Sequences

§2.10 Sums and Sequences

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§2.10(iii) Asymptotic Expansions of Entire Functions

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§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method

… ►See also Flajolet and Odlyzko (1990).

29: 2.3 Integrals of a Real Variable

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§2.3(i) Integration by Parts

… ►For the Fourier integral … ►

§2.3(iv) Method of Stationary Phase

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§2.3(v) Coalescing Peak and Endpoint: Bleistein’s Method

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§2.3(vi) Asymptotics of Mellin Transforms

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30: 13.8 Asymptotic Approximations for Large Parameters

§13.8 Asymptotic Approximations for Large Parameters

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§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

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§13.8(iii) Large $a$

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§13.8(iv) Large $a$ and $b$

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