# exceptional values

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## 1—10 of 55 matching pages

##### 1: 28.7 Analytic Continuation of Eigenvalues

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►The branch points are called the

*exceptional values*, and the other points*normal values*. The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …##### 2: 28.11 Expansions in Series of Mathieu Functions

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►See Meixner and Schäfke (1954, §2.28), and for expansions in the case of the exceptional values of $q$ see Meixner et al. (1980, p. 33).
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##### 3: 5.12 Beta Function

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►In this section all fractional powers have their principal values, except where noted otherwise.
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##### 4: 33.19 Power-Series Expansions in $r$

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►The expansions (33.19.1) and (33.19.3) converge for all finite values of $r$, except
$r=0$ in the case of (33.19.3).

##### 5: 10.72 Mathematical Applications

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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 ${z}_{0}(a)=0$.
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##### 6: 8.2 Definitions and Basic Properties

##### 7: 14.21 Definitions and Basic Properties

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►
${P}_{\nu}^{\pm \mu}\left(z\right)$ and ${\bm{Q}}_{\nu}^{\mu}\left(z\right)$ exist for all values of $\nu $, $\mu $, and $z$, except possibly $z=\pm 1$ and $\mathrm{\infty}$, which are branch points (or poles) of the functions, in general.
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##### 8: 1.4 Calculus of One Variable

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►For the functions discussed in the following DLMF chapters these two integration measures are adequate, as these special functions are

*analytic functions*of their variables, and thus ${C}^{\mathrm{\infty}}$, and well defined for all values of these variables; possible exceptions being at boundary points. …##### 9: 3.8 Nonlinear Equations

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►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$.
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##### 10: 1.9 Calculus of a Complex Variable

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►It is single-valued on $\u2102\setminus \{0\}$, except on the interval $(-\mathrm{\infty},0)$ where it is discontinuous and two-valued.
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