exceptional values

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1: 28.7 Analytic Continuation of Eigenvalues

… ►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

… ►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). …

3: 5.12 Beta Function

… ►In this section all fractional powers have their principal values, except where noted otherwise. …

4: 33.19 Power-Series Expansions in $r$

… ►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

… ►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 $z_0(a)=0$. …

6: 8.2 Definitions and Basic Properties

… ►Except where indicated otherwise in the DLMF these principal values are assumed. …

7: 14.21 Definitions and Basic Properties

… ► $P_{\nu }^{\pm \mu }\left(z\right)$ and ${𝑸}_{\nu }^{\mu }\left(z\right)$ 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. …

8: 1.4 Calculus of One Variable

… ►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

… ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. …

10: 1.9 Calculus of a Complex Variable

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