# asymptotic behavior of coefficients

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##### 2: Bibliography W
• G. Wolf (2008) On the asymptotic behavior of the Fourier coefficients of Mathieu functions. J. Res. Nat. Inst. Standards Tech. 113 (1), pp. 11–15.
• ##### 3: Bibliography J
• W. B. Jones and W. Van Assche (1998) 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.
• ##### 4: 2.10 Sums and Sequences
What is the asymptotic behavior of $f_{n}$ as $n\to\infty$ or $n\to-\infty$? More specially, what is the behavior of the higher coefficients in a Taylor-series expansion? …
• (c)

The coefficients in the Laurent expansion

2.10.27 $g(z)=\sum_{n=-\infty}^{\infty}g_{n}z^{n},$ $0<|z|,

have known asymptotic behavior as $n\to\pm\infty$.

• ##### 5: 1.8 Fourier Series
The series (1.8.1) is called the Fourier series of $f(x)$, and $a_{n},b_{n}$ are the Fourier coefficients of $f(x)$. …
###### Asymptotic Estimates of Coefficients
If $f(x)$ and $g(x)$ are continuous, have the same period and same Fourier coefficients, then $f(x)=g(x)$ for all $x$.
##### 7: 2.4 Contour Integrals
###### §2.4(i) Watson’s Lemma
with known asymptotic behavior as $t\to+\infty$. …For examples see Olver (1997b, pp. 315–320). … For integral representations of the $b_{2s}$ and their asymptotic behavior as $s\to\infty$ see Boyd (1995). … For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …
##### 8: 8.12 Uniform Asymptotic Expansions for Large Parameter
###### §8.12 Uniform Asymptotic Expansions for Large Parameter
where $g_{k}$, $k=0,1,2,\dots$, are the coefficients that appear in the asymptotic expansion (5.11.3) of $\Gamma\left(z\right)$. …where $d_{0,0}=-\tfrac{1}{3}$, … For the asymptotic behavior of $c_{k}(\eta)$ as $k\to\infty$ see Dunster et al. (1998) and Olde Daalhuis (1998c). …
##### 9: 30.9 Asymptotic Approximations and Expansions
###### §30.9(i) Prolate Spheroidal Wave Functions
The asymptotic behavior of $\lambda^{m}_{n}\left(\gamma^{2}\right)$ and $a^{m}_{n,k}(\gamma^{2})$ as $n\to\infty$ in descending powers of $2n+1$ is derived in Meixner (1944). …The asymptotic behavior of $\mathsf{Ps}^{m}_{n}\left(x,\gamma^{2}\right)$ and $\mathsf{Qs}^{m}_{n}\left(x,\gamma^{2}\right)$ as $x\to\pm 1$ is given in Erdélyi et al. (1955, p. 151). …
##### 10: 3.6 Linear Difference Equations
3.6.1 $a_{n}w_{n+1}-b_{n}w_{n}+c_{n}w_{n-1}=d_{n},$
The values of $w_{N}$ and $w_{N+1}$ needed to begin the backward recursion may be available, for example, from asymptotic expansions (§2.9). … A new problem arises, however, if, as $n\to\infty$, 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). … 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. … Here $\ell\in[0,k]$, and its actual value depends on the asymptotic behavior of the wanted solution in relation to those of the other solutions. …