asymptotic forms of higher coefficients

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1: 28.4 Fourier Series

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§28.4(vii) Asymptotic Forms for Large $m$

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2: 2.9 Difference Equations

… ►Formal solutions are … ► $c_0=1$, and higher coefficients are determined by formal substitution. … ►with $a_{0,j}=1$ and higher coefficients given by (2.9.7) (in the present case the coefficients of $a_{s,j}$ and $a_{s-1,j}$ are zero). … ►For analogous results for difference equations of the form … ►

3: 28.15 Expansions for Small $q$

§28.15 Expansions for Small $q$

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28.15.1 $${\lambda }_{\nu }\left(q\right)={\nu }^2+\frac{1}{2({\nu }^2-1)}{q}^2+\frac{5{\nu }^2+7}{32{({\nu }^2-1)}^3({\nu }^2-4)}{q}^4+\frac{9{\nu }^4+58{\nu }^2+29}{64{({\nu }^2-1)}^5({\nu }^2-4)({\nu }^2-9)}{q}^6+\mathrm{\cdots }.$$

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $q=h^2$: parameter and $\nu$: complex parameter

A&S Ref:

20.3.15 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

►Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$: …

4: 8.11 Asymptotic Approximations and Expansions

§8.11 Asymptotic Approximations and Expansions

… ►For an exponentially-improved asymptotic expansion (§2.11(iii)) see Olver (1991a). … ►where … ►See Tricomi (1950b) for these approximations, together with higher terms and extensions to complex variables. … ►

5: 2.11 Remainder Terms; Stokes Phenomenon

… ►with … ►For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004). … ►For higher-order differential equations, see Olde Daalhuis (1998a, b). … ►Furthermore, on proceeding to higher values of $n$ with higher precision, much more accuracy is achievable. … ►Their extrapolation is based on assumed forms of remainder terms that may not always be appropriate for asymptotic expansions. …

6: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►For examples see Olver (1997b, pp. 315–320). … ►The final expansion then has the form … ►Higher coefficients $b_{2s}$ in (2.4.15) can be found from (2.3.18) with $\lambda =1$, $\mu =2$, and $s$ replaced by $2s$. … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …

7: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

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§9.7(iii) Error Bounds for Real Variables

… ►Corresponding bounds for the errors in (9.7.7) to (9.7.14) may be obtained by use of these results and those of §9.2(v) and their differentiated forms. … ►For higher re-expansions of the remainder terms see Olde Daalhuis (1995, 1996), and Olde Daalhuis and Olver (1995a).

8: 3.6 Linear Difference Equations

… ►Many special functions satisfy second-order recurrence relations, or difference equations, of the form … ►The values of $w_N$ and $w_{N+1}$ needed to begin the backward recursion may be available, for example, from asymptotic expansions (§2.9). … ►The process is then repeated with a higher value of $N$, and the normalized solutions compared. … ►The normalizing factor $\mathrm{\Lambda }$ can be the true value of $w_0$ divided by its trial value, or $\mathrm{\Lambda }$ can be chosen to satisfy a known property of the wanted solution of the form … ►For further information, including a more general form of normalizing condition, other examples, convergence proofs, and error analyses, see Olver (1967a), Olver and Sookne (1972), and Wimp (1984, Chapter 6). …

9: Bibliography R

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A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math. 33 (4), pp. 915–928.

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External Links:

ISSN 0008-414X, Document, Link, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.18(v).

Encodings:

BibTeX

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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.

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External Links:

MathReview (T. W. Kao), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.

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External Links:

ISBN 5-7966-0570-4, MathReview (Guillermo López Lagomasino), ZentralBlatt

Referenced by:

§12.11(ii), §6.13.

Encodings:

BibTeX

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Functional Analysis. McGraw-Hill Book Co., New York.

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Notes:

McGraw-Hill Series in Higher Mathematics

External Links:

MathReview (F. Smithies), ZentralBlatt

Referenced by:

§1.18(x), §2.6(i).

Encodings:

BibTeX

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On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.

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External Links:

ISBN 3-7643-6287-1, MathReview Entry, ZentralBlatt

Referenced by:

5th item.

Encodings:

BibTeX

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10: Bibliography

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On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.

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External Links:

ISSN 0065-1036, Pdf, MathReview (L. Skula), ZentralBlatt

Referenced by:

§24.12(iii).

Encodings:

BibTeX

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Formulas for higher derivatives of the Riemann zeta function. Math. Comp. 44 (169), pp. 223–232.

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External Links:

ISSN 0025-5718, FullText, MathReview (Aleksandar Ivić), ZentralBlatt

Referenced by:

(25.11.19), (25.11.20), (25.11.6), §25.11(vi), §25.18(i), (25.4.5), (25.4.6), §25.4, (25.6.11), (25.6.12), (25.6.13), (25.6.14), §25.6(ii), §25.6.

Encodings:

BibTeX

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Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.

External Links:

Document, MathReview (D. Siersma), ZentralBlatt

Referenced by:

§36.2(i), Ch.36.

Encodings:

BibTeX

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Matching higher conserved charges for strings and spins. J. High Energy Phys. 2004 (3).

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Notes:

(electronic only, article id. JHEP03(2004)004, 32 pp.).

External Links:

ISSN 1126-6708, Document, MathReview

Referenced by:

§19.35(ii).

Encodings:

BibTeX

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Commentary on the Paper “Beiträge zur Theorie der Toeplitzschen Form”. In Gábor Szegő, Collected Papers. Vol. 1, Contemporary Mathematicians, pp. 303–305.

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External Links:

ISBN 3-7643-3056-2, MathReview (G. Piranian)

Referenced by:

§18.33(iv), §18.33(iv), §18.33(v).

Encodings:

BibTeX

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