asymptotic forms of higher coefficients
(0.004 seconds)
1—10 of 13 matching pages
1: 28.4 Fourier Series
…
►
§28.4(vii) Asymptotic Forms for Large
…2: 2.9 Difference Equations
…
►Formal solutions are
…
►
, and higher coefficients are determined by formal substitution.
…
►with and higher coefficients given by (2.9.7) (in the present case the coefficients of and are zero).
…
►For analogous results for difference equations of the form
…
►
3: 28.15 Expansions for Small
§28.15 Expansions for Small
… ►
28.15.1
►Higher coefficients can be found by equating powers of in the following continued-fraction equation, with :
…
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 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
…
►
§2.4(i) Watson’s Lemma
… ►For examples see Olver (1997b, pp. 315–320). … ►The final expansion then has the form … ►Higher coefficients in (2.4.15) can be found from (2.3.18) with , , and replaced by . … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …7: Bibliography R
…
►
Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow.
Studies in Appl. Math. 53, pp. 91–110.
►
Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory.
Studies in Appl. Math. 53, pp. 217–224.
…
►
Asymptotics and Bounds of the Roots of Equations (Russian).
Zinatne, Riga.
…
►
Functional Analysis.
McGraw-Hill Book Co., New York.
…
►
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.
…
8: 9.7 Asymptotic Expansions
§9.7 Asymptotic Expansions
… ►Also and for , … ►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 and 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 , and the normalized solutions compared.
…
►The normalizing factor can be the true value of divided by its trial value, or 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
…
►
Intégrandes à deux formes quadratiques.
C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
…
►
Asymptotics and closed form of a generalized incomplete gamma function.
J. Comput. Appl. Math. 67 (2), pp. 371–379.
…
►
Asymptotics of Racah coefficients and polynomials.
J. Phys. A 32 (3), pp. 537–553.
…
►
Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges.
6th edition, Vol. 1, Chelsea Publishing Co., New York.
►
Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges.
6th edition, Vol. 2, Chelsea Publishing Co., New York.
…