asymptotic%20behavior%20of%20coefficients
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11: 2.11 Remainder Terms; Stokes Phenomenon
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§2.11(i) Numerical Use of Asymptotic Expansions
… ►Secondly, the asymptotic series represents an infinite class of functions, and the remainder depends on which member we have in mind. … ► … ► … ►For example, using double precision is found to agree with (2.11.31) to 13D. …12: 27.2 Functions
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►Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►Gauss and Legendre conjectured that is asymptotic to as :
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27.2.3
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►An equivalent form states that the th prime (when the primes are listed in increasing order) is asymptotic to as :
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27.2.4
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13: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
… ►When the zeros are asymptotically given by and , where is a large positive integer and … ►§12.11(iii) Asymptotic Expansions for Large Parameter
►For large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …The first two coefficients are given by …14: Bibliography F
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Sur certaines sommes des intégral-cosinus.
Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).
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Tables of Elliptic Integrals of the First, Second, and Third Kind.
Technical report
Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.
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On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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On weighted polynomial approximation on the whole real axis.
Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.
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15: 5.11 Asymptotic Expansions
§5.11 Asymptotic Expansions
… ►and … ►Wrench (1968) gives exact values of up to . … ►§5.11(iii) Ratios
… ►16: 26.5 Lattice Paths: Catalan Numbers
17: 25.12 Polylogarithms
18: Bibliography O
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Asymptotic Enumeration Methods.
In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.),
pp. 1063–1229.
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On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles.
Methods Appl. Anal. 7 (4), pp. 727–745.
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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series.
J. Inst. Math. Appl. 20 (3), pp. 379–391.
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A paradox in asymptotics.
SIAM J. Math. Anal. 1 (4), pp. 533–534.
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Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations.
Methods Appl. Anal. 1 (1), pp. 1–13.
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19: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►§28.8(ii) Sips’ Expansions
… ►§28.8(iii) Goldstein’s Expansions
… ►Barrett’s Expansions
… ►20: Bibliography C
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Asymptotic estimates for generalized Stirling numbers.
Analysis (Munich) 20 (1), pp. 1–13.
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Asymptotics of Racah coefficients and polynomials.
J. Phys. A 32 (3), pp. 537–553.
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Validated computation of certain hypergeometric functions.
ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.
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Coulomb effects in the Klein-Gordon equation for pions.
Phys. Rev. C 20 (2), pp. 696–704.
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Correlation between pole location and asymptotic behavior for Painlevé I solutions.
Comm. Pure Appl. Math. 52 (4), pp. 461–478.
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