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11: 3.10 Continued Fractions

… ► $C_n$ is the $n$th approximant or convergent to $C$. … ►Every convergent, asymptotic, or formal series … ►However, other continued fractions with the same limit may converge in a much larger domain of the complex plane than the fraction given by (3.10.4) and (3.10.5). … ►For the same function $f(z)$, the convergent $C_n$ of the Jacobi fraction (3.10.11) equals the convergent $C_{2n}$ of the Stieltjes fraction (3.10.6). … ►For further information on the preceding algorithms, including convergence in the complex plane and methods for accelerating convergence, see Blanch (1964) and Lorentzen and Waadeland (1992, Chapter 3). …

12: 1.10 Functions of a Complex Variable

… ►The series (1.10.6) converges uniformly and absolutely on compact sets in the annulus. … ►

§1.10(ix) Infinite Products

… ►The convergence of the infinite product is uniform if the sequence of partial products converges uniformly. ►

$M$-test

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13: 8.25 Methods of Computation

… ►Although the series expansions in §§8.7, 8.19(iv), and 8.21(vi) converge for all finite values of $z$, they are cumbersome to use when $|z|$ is large owing to slowness of convergence and cancellation. …

14: 14.32 Methods of Computation

… ►In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. …

15: 15.19 Methods of Computation

… ►The Gauss series (15.2.1) converges for . … ►However, by appropriate choice of the constant $z_0$ in (15.15.1) we can obtain an infinite series that converges on a disk containing $z={\mathrm{e}}^{\pm \pi \mathrm{i}/3}$. Moreover, it is also possible to accelerate convergence by appropriate choice of $z_0$. ►Large values of $|a|$ or $|b|$, for example, delay convergence of the Gauss series, and may also lead to severe cancellation. …

16: 13.5 Continued Fractions

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side everywhere in $ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

17: 13.17 Continued Fractions

… ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side for all $z\in ℂ$. For more details on how a continued fraction converges to a meromorphic function see Jones and Thron (1980). … ►This continued fraction converges to the meromorphic function of $z$ on the left-hand side throughout the sector . …

18: 6.18 Methods of Computation

… ►For large $x$ or $|z|$ these series suffer from slow convergence or cancellation (or both). However, this problem is less severe for the series of spherical Bessel functions because of their more rapid rate of convergence, and also (except in the case of (6.10.6)) absence of cancellation when $z=x$ ($>0$). … ►Convergence becomes slow when $z$ is near the negative real axis, however. …

19: 33.8 Continued Fractions

… ►The continued fraction (33.8.1) converges for all finite values of $\rho$, and (33.8.2) converges for all $\rho \ne 0$. … ►The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision. …

20: 15.15 Sums

… ►Here $z_0$ ($\ne 0$) is an arbitrary complex constant and the expansion converges when $|z-z_0|>\max (|z_0|,|z_0-1|)$. …