# asymptotic behavior

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## 1—10 of 32 matching pages

##### 1: 30.9 Asymptotic Approximations and Expansions

###### §30.9 Asymptotic Approximations and Expansions

►###### §30.9(i) Prolate Spheroidal Wave Functions

… ►The asymptotic behavior of ${\lambda}_{n}^{m}\left({\gamma}^{2}\right)$ and ${a}_{n,k}^{m}({\gamma}^{2})$ as $n\to \mathrm{\infty}$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of ${\mathsf{Ps}}_{n}^{m}(x,{\gamma}^{2})$ and ${\mathsf{Qs}}_{n}^{m}(x,{\gamma}^{2})$ as $x\to \pm 1$ is given in Erdélyi et al. (1955, p. 151). …##### 2: 18.16 Zeros

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###### Asymptotic Behavior

… ►###### Asymptotic Behavior

… ►For an asymptotic expansion of ${x}_{n,m}$ as $n\to \mathrm{\infty}$ that applies uniformly for $m=1,2,\mathrm{\dots},\lfloor \frac{1}{2}n\rfloor $, see Olver (1959, §14(i)). …##### 3: Bibliography J

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Asymptotic behavior of integrals.
SIAM Rev. 14 (2), pp. 286–317.
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Asymptotic behavior of the continued fraction coefficients of a class of Stieltjes transforms including the Binet function.
In Orthogonal functions, moment theory, and continued fractions
(Campinas, 1996),
Lecture Notes in Pure and Appl. Math., Vol. 199, pp. 257–274.
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##### 4: 19.12 Asymptotic Approximations

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►With $\psi \left(x\right)$ denoting the digamma function (§5.2(i)) in this subsection, the asymptotic behavior of $K\left(k\right)$ and $E\left(k\right)$ near the singularity at $k=1$ is given by the following convergent series:
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►For the asymptotic behavior of $F(\varphi ,k)$ and $E(\varphi ,k)$ as $\varphi \to \frac{1}{2}\pi -$ and $k\to 1-$ see Kaplan (1948, §2), Van de Vel (1969), and Karp and Sitnik (2007).
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##### 5: 33.23 Methods of Computation

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►Use of extended-precision arithmetic increases the radial range that yields accurate results, but eventually other methods must be employed, for example, the asymptotic expansions of §§33.11 and 33.21.
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►On the other hand, the irregular solutions of §§33.2(iii) and 33.14(iii) need to be integrated in the direction of decreasing radii beginning, for example, with values obtained from asymptotic expansions (§§33.11 and 33.21).
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►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions.
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##### 6: 19.10 Relations to Other Functions

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►In each case when $y=1$, the quantity multiplying ${R}_{C}$ supplies the asymptotic behavior of the left-hand side as the left-hand side tends to 0.
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##### 7: 30.11 Radial Spheroidal Wave Functions

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###### §30.11(iii) Asymptotic Behavior

…##### 8: 3.6 Linear Difference Equations

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►A new problem arises, however, if, as $n\to \mathrm{\infty}$, the asymptotic behavior of ${w}_{n}$ is intermediate to those of two independent solutions ${f}_{n}$ and ${g}_{n}$ of the corresponding inhomogeneous equation (the complementary functions).
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►Thus the asymptotic behavior of the particular solution ${\mathbf{E}}_{n}\left(1\right)$ is intermediate to those of the complementary functions ${J}_{n}\left(1\right)$ and ${Y}_{n}\left(1\right)$; moreover, the conditions for Olver’s algorithm are satisfied.
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►Here $\mathrm{\ell}\in [0,k]$, and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions.
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##### 9: Bibliography W

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Asymptotic behavior of stability regions for Hill’s equation.
SIAM J. Appl. Math. 47 (5), pp. 941–958.
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On the asymptotic behavior of the Fourier coefficients of Mathieu functions.
J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
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##### 10: 1.8 Fourier Series

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