asymptotic%20behavior%20of%20coefficients
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21: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
►§11.6(i) Large , Fixed
… ►§11.6(ii) Large , Fixed
… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …22: 8 Incomplete Gamma and Related
Functions
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23: 28 Mathieu Functions and Hill’s Equation
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24: Bibliography D
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Recherches analytiques sur la théorie des nombres premiers. Première partie. La fonction de Riemann et les nombres premiers en général, suivi d’un Appendice sur des réflexions applicables à une formule donnée par Riemann.
Ann. Soc. Sci. Bruxelles 20, pp. 183–256 (French).
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Recherches analytiques sur la théorie des nombres premiers. Deuxième partie. Les fonctions de Dirichlet et les nombres premiers de la forme linéaire
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Ann. Soc. Sci. Bruxelles 20, pp. 281–397 (French).
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Complex zeros of cylinder functions.
Math. Comp. 20 (94), pp. 215–222.
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On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function.
Methods Appl. Anal. 5 (3), pp. 223–247.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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25: 2.2 Transcendental Equations
§2.2 Transcendental Equations
… ►An important case is the reversion of asymptotic expansions for zeros of special functions. In place of (2.2.1) assume that …where and () is the coefficient of in the asymptotic expansion of (Lagrange’s formula for the reversion of series). …For other examples see de Bruijn (1961, Chapter 2).26: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10 Uniform Asymptotic Expansions for Large Parameter
… ►Lastly, the function in (12.10.3) and (12.10.4) has the asymptotic expansion: … ►The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). … ►27: 8.26 Tables
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Khamis (1965) tabulates for , to 10D.
Abramowitz and Stegun (1964, pp. 245–248) tabulates for , to 7D; also for , to 6S.
Pagurova (1961) tabulates for , to 4-9S; for , to 7D; for , to 7S or 7D.
Zhang and Jin (1996, Table 19.1) tabulates for , to 7D or 8S.
28: 23 Weierstrass Elliptic and Modular
Functions
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29: 25.11 Hurwitz Zeta Function
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§25.11(xii) -Asymptotic Behavior
… ►As in the sector , with and fixed, we have the asymptotic expansion … ►Similarly, as in the sector , ►
25.11.44
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30: 3.8 Nonlinear Equations
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►However, when the coefficients are all real, complex arithmetic can be avoided by the following iterative process.
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►Initial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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►Thus if is the polynomial (3.8.8) and is the coefficient
, say, then
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►Consider and .
We have and .
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