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asymptotic forms of higher coefficients

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1: 28.4 Fourier Series
§28.4(vii) Asymptotic Forms for Large m
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
28.15.1 λ ν ( q ) = ν 2 + 1 2 ( ν 2 1 ) q 2 + 5 ν 2 + 7 32 ( ν 2 1 ) 3 ( ν 2 4 ) q 4 + 9 ν 4 + 58 ν 2 + 29 64 ( ν 2 1 ) 5 ( ν 2 4 ) ( ν 2 9 ) q 6 + .
Higher coefficients can be found by equating powers of q in the following continued-fraction equation, with a = λ ν ( q ) : …
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
§2.4(i) Watson’s Lemma
For examples see Olver (1997b, pp. 315–320). … The final expansion then has the formHigher coefficients b 2 s in (2.4.15) can be found from (2.3.18) with λ = 1 , μ = 2 , and s replaced by 2 s . … For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …
7: Bibliography R
  • W. H. Reid (1974a) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. I. Plane Couette flow. Studies in Appl. Math. 53, pp. 91–110.
  • W. H. Reid (1974b) Uniform asymptotic approximations to the solutions of the Orr-Sommerfeld equation. II. The general theory. Studies in Appl. Math. 53, pp. 217–224.
  • È. Ya. Riekstynš (1991) Asymptotics and Bounds of the Roots of Equations (Russian). Zinatne, Riga.
  • W. Rudin (1973) Functional Analysis. McGraw-Hill Book Co., New York.
  • J. Rushchitsky and S. Rushchitska (2000) 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 u 0 = v 0 = 1 and for k = 1 , 2 , , … 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 formThe 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 Λ can be the true value of w 0 divided by its trial value, or Λ can be chosen to satisfy a known property of the wanted solution of the formFor 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
  • B. C. Carlson (1972b) Intégrandes à deux formes quadratiques. C. R. Acad. Sci. Paris Sér. A–B 274 (15 May, 1972, Sér. A), pp. 1458–1461 (French).
  • M. A. Chaudhry, N. M. Temme, and E. J. M. Veling (1996) Asymptotics and closed form of a generalized incomplete gamma function. J. Comput. Appl. Math. 67 (2), pp. 371–379.
  • L. Chen, M. E. H. Ismail, and P. Simeonov (1999) Asymptotics of Racah coefficients and polynomials. J. Phys. A 32 (3), pp. 537–553.
  • G. Chrystal (1959a) Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 1, Chelsea Publishing Co., New York.
  • G. Chrystal (1959b) Algebra: An Elementary Textbook for the Higher Classes of Secondary Schools and for Colleges. 6th edition, Vol. 2, Chelsea Publishing Co., New York.