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
…2: 2.9 Difference Equations
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►Formal solutions are
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, and higher coefficients are determined by formal substitution.
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►with and higher coefficients given by (2.9.7) (in the present case the coefficients of and 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
§28.15 Expansions for Small
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28.15.1
►Higher coefficients can be found by equating powers of in the following continued-fraction equation, with :
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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 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 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: 9.7 Asymptotic Expansions
§9.7 Asymptotic Expansions
… ►§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
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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 and 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 , and the normalized solutions compared.
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►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
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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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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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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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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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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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Formulas for higher derivatives of the Riemann zeta function.
Math. Comp. 44 (169), pp. 223–232.
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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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Matching higher conserved charges for strings and spins.
J. High Energy Phys. 2004 (3).
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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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