# analytically continued

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## 11—20 of 38 matching pages

##### 11: Bibliography W

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Analytic Theory of Continued Fractions.
D. Van Nostrand Company, Inc., New York.
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##### 12: 8.17 Incomplete Beta Functions

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►However, in the case of §8.17 it is straightforward to continue most results analytically to other real values of $a$, $b$, and $x$, and also to complex values.
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##### 13: 1.12 Continued Fractions

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►For analytical and numerical applications of continued fractions to special functions see §3.10.
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##### 14: 3.8 Nonlinear Equations

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►This is an iterative method for real twice-continuously differentiable, or complex analytic, functions:
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##### 15: 33.23 Methods of Computation

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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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##### 16: 4.37 Inverse Hyperbolic Functions

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4.37.6
$$\mathrm{Arccoth}z=\mathrm{Arctanh}\left(1/z\right).$$

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►Elsewhere on the integration paths in (4.37.1) and (4.37.2) the branches are determined by continuity.
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►These functions are analytic in the cut plane depicted in Figure 4.37.1(iv), (v), (vi), respectively.
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►It should be noted that the imaginary axis is not a cut; the function defined by (4.37.19) and (4.37.20) is analytic everywhere except on $(-\mathrm{\infty},1]$.
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##### 17: 4.12 Generalized Logarithms and Exponentials

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►Both $\varphi (x)$ and $\psi (x)$ are continuously differentiable.
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►For analytic generalized logarithms, see Kneser (1950).

##### 18: Frank W. J. Olver

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►Olver continued to maintain a connection to NIST after moving to the university.
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►He continued his editing work until the time of his death on April 22, 2013 at age 88.

##### 19: 10.72 Mathematical Applications

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►In regions in which (10.72.1) has a simple turning point ${z}_{0}$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and ${z}_{0}$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)).
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►The number $m$ can also be replaced by any real constant $\lambda $
$(>-2)$ in the sense that ${(z-{z}_{0})}^{-\lambda}$
$f(z)$ is analytic and nonvanishing at ${z}_{0}$; moreover, $g(z)$ is permitted to have a single or double pole at ${z}_{0}$.
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►In regions in which the function $f(z)$ has a simple pole at $z={z}_{0}$ and ${(z-{z}_{0})}^{2}g(z)$ is analytic at $z={z}_{0}$ (the case $\lambda =-1$ in §10.72(i)), asymptotic expansions of the solutions $w$ of (10.72.1) for large $u$ can be constructed in terms of Bessel functions and modified Bessel functions of order $\pm \sqrt{1+4\rho}$, where $\rho $ is the limiting value of ${(z-{z}_{0})}^{2}g(z)$ as $z\to {z}_{0}$.
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►In (10.72.1) assume $f(z)=f(z,\alpha )$ and $g(z)=g(z,\alpha )$ depend continuously on a real parameter $\alpha $, $f(z,\alpha )$ has a simple zero $z={z}_{0}(\alpha )$ and a double pole $z=0$, except for a critical value $\alpha =a$, where ${z}_{0}(a)=0$.
Assume that whether or not $\alpha =a$, ${z}^{2}g(z,\alpha )$ is analytic at $z=0$.
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##### 20: 31.11 Expansions in Series of Hypergeometric Functions

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►For example, consider the Heun function which is analytic at $z=a$ and has exponent $\alpha $ at $\mathrm{\infty}$.
…In this case the accessory parameter $q$ is a root of the continued-fraction equation
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