asymptotic approximations of sums and sequences
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1: 2.10 Sums and Sequences
§2.10 Sums and Sequences
… ► … ►§2.10(iii) Asymptotic Expansions of Entire Functions
… ►§2.10(iv) Taylor and Laurent Coefficients: Darboux’s Method
… ►See also Flajolet and Odlyzko (1990).2: 2.1 Definitions and Elementary Properties
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►means that for each , the difference between and the th partial sum on the right-hand side is as in .
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§2.1(iv) Uniform Asymptotic Expansions
… ►§2.1(v) Generalized Asymptotic Expansions
… ►Then is an asymptotic sequence or scale. Suppose also that and satisfy …3: 5.4 Special Values and Extrema
4: Bibliography W
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Asymptotic approximations for certain - and - symbols.
J. Phys. A 32 (39), pp. 6901–6902.
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Computation of the Whittaker function of the second kind by summing its divergent asymptotic series with the help of nonlinear sequence transformations.
Computers in Physics 10 (5), pp. 496–503.
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Asymptotic Approximations to Truncation Errors of Series Representations for Special Functions.
In Algorithms for Approximation, A. Iske and J. Levesley (Eds.),
pp. 331–348.
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On uniform asymptotic expansion of definite integrals.
J. Approximation Theory 7 (1), pp. 76–86.
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Error bounds for asymptotic approximations of special functions.
Ann. Numer. Math. 2 (1-4), pp. 181–197.
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5: 4.45 Methods of Computation
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►Beginning with , generate the sequence
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►Another method, when is large, is to sum
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►Initial approximations are obtainable, for example, from the power series (4.13.6) (with ) when is close to , from the asymptotic expansion (4.13.10) when is large, and by numerical integration of the differential equation (4.13.4) (§3.7) for other values of .
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6: Bibliography C
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Asymptotic approximations for symmetric elliptic integrals.
SIAM J. Math. Anal. 25 (2), pp. 288–303.
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Introduction to Approximation Theory.
2nd edition, Chelsea Publishing Co., New York.
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Algorithm 597: Sequence of modified Bessel functions of the first kind.
ACM Trans. Math. Software 9 (2), pp. 242–245.
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Department of Computer Science, University of Victoria, Canada.
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A sequence of series for the Lambert function.
In Proceedings of the 1997 International Symposium on
Symbolic and Algebraic Computation (Kihei, HI),
pp. 197–204.
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7: Bibliography R
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Rational Chebyshev approximation by Remes’ algorithms.
Numer. Math. 7 (4), pp. 322–330.
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Remark on Algorithm 498: Airy functions using Chebyshev series approximations.
ACM Trans. Math. Software 7 (3), pp. 404–405.
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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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Research Institute for Symbolic Computation, Hagenberg im Mühlkreis, Austria.
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8: Bibliography S
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Calculator function approximation.
Amer. Math. Monthly 90 (5), pp. 317–325.
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The accuracy of iterated JWBK approximations for Coulomb radial functions.
Comput. Phys. Comm. 32 (2), pp. 115–119.
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Non-linear transformations of divergent and slowly convergent sequences.
J. Math. Phys. 34, pp. 1–42.
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The On-Line Encyclopedia of Integer Sequences.
Notices Amer. Math. Soc. 50 (8), pp. 912–915.
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Approximate Calculation of Multiple Integrals.
Prentice-Hall Inc., Englewood Cliffs, N.J..
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9: Bibliography L
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Asymptotic Analysis.
Mathematical Centre Tracts, Mathematisch Centrum, Amsterdam.
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Monotonic sequences related to zeros of Bessel functions.
Numer. Algorithms 49 (1-4), pp. 221–233.
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Approximations for elliptic integrals.
Math. Comp. 22 (103), pp. 627–634.
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Further approximations for elliptic integrals.
Math. Comp. 24 (109), pp. 191–198.
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Algorithms for rational approximations for a confluent hypergeometric function.
Utilitas Math. 11, pp. 123–151.
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10: Bibliography K
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Unimodal sequences show that Lambert is Bernstein.
C. R. Math. Acad. Sci. Soc. R. Can. 33 (2), pp. 50–56.
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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity.
J. Comput. Appl. Math. 205 (1), pp. 186–206.
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Uniform asymptotic approximations for the Meixner-Sobolev polynomials.
Anal. Appl. (Singap.) 10 (3), pp. 345–361.
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Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2153), pp. 20130008, 12.
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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters.
J. B. Wolters, Groningen.
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