asymptotic approximation of integrals

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1: 36.11 Leading-Order Asymptotics

§36.11 Leading-Order Asymptotics

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2: 36.12 Uniform Approximation of Integrals

… ►The canonical integrals (36.2.4) provide a basis for uniform asymptotic approximations of oscillatory integrals. … ►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).

3: 19.38 Approximations

§19.38 Approximations

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4: 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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5: 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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6: 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). …

7: 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). …

8: 19.27 Asymptotic Approximations and Expansions

§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)). … ►

9: Roderick S. C. Wong

… ►He is the author of the book Asymptotic Approximations of Integrals, published by Academic Press in 1989 and reprinted by SIAM in its Classics in Applied Mathematics Series in 2001, and of Lecture Notes on Applied Analysis, published by World Scientific in 2010. …

10: 2.2 Transcendental Equations

… ►For other examples see de Bruijn (1961, Chapter 2).