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1: 5.11 Asymptotic Expansions

… ►The expansion (5.11.1) is called Stirling’s series (Whittaker and Watson (1927, §12.33)), whereas the expansion (5.11.3), or sometimes just its leading term, is known as Stirling’s formula (Abramowitz and Stegun (1964, §6.1), Olver (1997b, p. 88)). … ►If the sums in the expansions (5.11.1) and (5.11.2) are terminated at $k=n-1$ ($k\ge 0$) and $z$ is real and positive, then the remainder terms are bounded in magnitude by the first neglected terms and have the same sign. … ►For the remainder term in (5.11.3) write … ►For re-expansions of the remainder terms in (5.11.1) and (5.11.3) in series of incomplete gamma functions with exponential improvement (§2.11(iii)) in the asymptotic expansions, see Berry (1991), Boyd (1994), and Paris and Kaminski (2001, §6.4). … ►For the error term in (5.11.19) in the case $z=x(>0)$ and $c=1$, see Olver (1995). …

2: 2.11 Remainder Terms; Stokes Phenomenon

… ►Taking $m=10$ in (2.11.2), the first three terms give us the approximation …The error term is, in fact, approximately 700 times the last term obtained in (2.11.4). … ►These answers are linked to the terms involving the complementary error function in the more powerful expansions typified by the combination of (2.11.10) and (2.11.15). … ►However, to enjoy the resurgence property (§2.7(ii)) we often seek instead expansions in terms of the $F$-functions introduced in §2.11(iii), leaving the connection of the error-function type behavior as an implicit consequence of this property of the $F$-functions. … ►Subtraction of this result from the sum of the first 5 terms in (2.11.25) yields 0. …

3: Bibliography O

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions. J. Soc. Indust. Appl. Math. Ser. B Numer. Anal. 2 (2), pp. 225–243.

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External Links:

Document, MathReview (T. E. Hull), ZentralBlatt

Referenced by:

§13.19, §13.7(ii).

Encodings:

BibTeX

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A paradox in asymptotics. SIAM J. Math. Anal. 1 (4), pp. 533–534.

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External Links:

Document, MathReview (L. Hoischen), ZentralBlatt

Referenced by:

§2.10(iv).

Encodings:

BibTeX

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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions. Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.

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External Links:

ISSN 0308-2105, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§13.20(iii), §13.20(iv).

Encodings:

BibTeX

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Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.

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Notes:

Reprinted in 2000 by the Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pennsylvania

External Links:

MathReview (J. F. Traub), ZentralBlatt (Nelli Dimitrova (Sofia))

Referenced by:

§3.8(vii).

Encodings:

BibTeX

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4: 18.39 Applications in the Physical Sciences

… ►The functions ${\varphi }_n$ are expressed in terms of Romanovski–Bessel polynomials, or Laguerre polynomials by (18.34.7_1). …The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by ${(\alpha \mathrm{\hslash }\gamma )}^2/(2m)$ ($\gamma \ge 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha (x-x_e)/2}{W}_{\lambda ,\mathrm{i}\gamma }\left(2\lambda {\mathrm{e}}^{-\alpha (x-x_e)}\right)$ expressed in terms of Whittaker functions (13.14.3). … ►

a) Spherical Radial Coulomb Wave Functions Expressed in terms of Laguerre OP’s

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d) Radial Coulomb Wave Functions Expressed in Terms of the Associated Coulomb–Laguerre OP’s

… ►which corresponds to the exact results, in terms of Whittaker functions, of §§33.2 and 33.14, in the sense that projections onto the functions ${\varphi }_{n,l}(sr)/r$, the functions bi-orthogonal to ${\varphi }_{n,l}(sr)$, are identical. …

5: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

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§12.11(ii) Asymptotic Expansions of Large Zeros

… ►Numerical calculations in this case show that $z_{\frac{1}{2},s}$ corresponds to the $s$th zero on the string; compare §7.13(ii). ►

§12.11(iii) Asymptotic Expansions for Large Parameter

… ►For further information, including associated functions, see Olver (1959).