# asymptotic forms of higher coefficients

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## 1—10 of 13 matching pages

##### 1: 28.4 Fourier Series

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

…##### 2: 2.9 Difference Equations

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►Formal solutions are
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${c}_{0}=1$, and higher coefficients are determined by formal substitution.
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►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).
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►For analogous results for difference equations of the form
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##### 3: 28.15 Expansions for Small $q$

###### §28.15 Expansions for Small $q$

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

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

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►with
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►For higher-order Stokes phenomena see Olde Daalhuis (2004b) and Howls et al. (2004).
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►For higher-order differential equations, see Olde Daalhuis (1998a, b).
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►Furthermore, on proceeding to higher values of $n$ with higher precision, much more accuracy is achievable.
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►Their extrapolation is based on assumed forms of remainder terms that may not always be appropriate for asymptotic expansions.
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##### 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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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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Asymptotics and Bounds of the Roots of Equations (Russian).
Zinatne, Riga.
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Functional Analysis.
McGraw-Hill Book Co., New York.
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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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##### 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

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►Many special functions satisfy second-order recurrence relations, or difference equations, of the form
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►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).
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►The process is then repeated with a higher value of $N$, and the normalized solutions compared.
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►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
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►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).
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##### 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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Asymptotics and closed form of a generalized incomplete gamma function.
J. Comput. Appl. Math. 67 (2), pp. 371–379.
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Asymptotics of Racah coefficients and polynomials.
J. Phys. A 32 (3), pp. 537–553.
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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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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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