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: Bibliography R

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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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Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.

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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:

§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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8: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

… ►Also $u_0=v_0=1$ and for $k=1,2,\mathrm{\dots }$, … ►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).

9: 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). …

10: Bibliography C

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Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).

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External Links:

MathReview (J. A. Donaldson), ZentralBlatt

Referenced by:

§19.31.

Encodings:

BibTeX

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Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.16.

Encodings:

BibTeX

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Asymptotics of Racah coefficients and polynomials. J. Phys. A 32 (3), pp. 537–553.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§34.8, §34.8.

Encodings:

BibTeX

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Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 1, Chelsea Publishing Co., New York.

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Notes:

A seventh edition (reprinting of the sixth edition) was published by AMS Chelsea Publishing Series, American Mathematical Society, 1964.

External Links:

ISBN 978-0-8218-1648-6, Seventh Edition, MathReview, ZentralBlatt

Referenced by:

§1.2(i), §1.2(ii), §1.2(iii), §1.2(iii).

Encodings:

BibTeX

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Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 2, Chelsea Publishing Co., New York.

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Notes:

A seventh edition (reprinting of the sixth edition) was published by AMS Chelsea Publishing Series, American Mathematical Society, 1964.

External Links:

ISBN 978-0-8218-1649-3, Seventh Edition, MathReview, ZentralBlatt

Referenced by:

§1.2(i).

Encodings:

BibTeX

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