asymptotic approximations and expansions

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11: 35.10 Methods of Computation

… ►For small values of $\Vert 𝐓\Vert$ the zonal polynomial expansion given by (35.8.1) can be summed numerically. For large $\Vert 𝐓\Vert$ the asymptotic approximations referred to in §35.7(iv) are available. … ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

12: 2.11 Remainder Terms; Stokes Phenomenon

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§2.11(i) Numerical Use of Asymptotic Expansions

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§2.11(iii) Exponentially-Improved Expansions

… ►For another approach see Paris (2001a, b). ►

§2.11(vi) Direct Numerical Transformations

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13: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). … ►Furthermore, as $t\to 0+$, $q(t)$ has the expansion (2.3.7). … ►For examples see Olver (1997b, pp. 315–320). ►

§2.4(iii) Laplace’s Method

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14: 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). ►The complementary error function also plays a ubiquitous role in constructing exponentially-improved asymptotic expansions and providing a smooth interpretation of the Stokes phenomenon; see §§2.11(iii) and 2.11(iv). … ►For applications in statistics and probability theory, also for the role of the normal distribution functions (the error functions and probability integrals) in the asymptotics of arbitrary probability density functions, see Johnson et al. (1994, Chapter 13) and Patel and Read (1982, Chapters 2 and 3).

15: 2.10 Sums and Sequences

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§2.10(iii) Asymptotic Expansions of Entire Functions

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16: 2.3 Integrals of a Real Variable

… ►For the Fourier integral … ►

§2.3(ii) Watson’s Lemma

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§2.3(iii) Laplace’s Method

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17: 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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18: 23.12 Asymptotic Approximations

§23.12 Asymptotic Approximations

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19: 19.38 Approximations

§19.38 Approximations

… ►Approximations of the same type for $K\left(k\right)$ and $E\left(k\right)$ for are given in Cody (1965a) with maximum absolute errors ranging from 4×10⁻⁵ to 4×10⁻¹⁸. Cody (1965b) gives Chebyshev-series expansions (§3.11(ii)) with maximum precision 25D. ►Approximations for Legendre’s complete or incomplete integrals of all three kinds, derived by Padé approximation of the square root in the integrand, are given in Luke (1968, 1970). …The accuracy is controlled by the number of terms retained in the approximation; for real variables the number of significant figures appears to be roughly twice the number of terms retained, perhaps even for $\varphi$ near $\pi /2$ with the improvements made in the 1970 reference. …

20: 28.8 Asymptotic Expansions for Large $q$

§28.8 Asymptotic Expansions for Large $q$

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Barrett’s Expansions

… ►With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). …