exceptional values

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11: 3.6 Linear Difference Equations

… ►or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …

12: 2.6 Distributional Methods

… ►We may therefore define the integral on the left-hand side of (2.6.4) by the value on the right-hand side, except when $\alpha ,\beta =0,-1,-2,\mathrm{\dots }$. …

13: 1.10 Functions of a Complex Variable

… ►In any neighborhood of an isolated essential singularity, however small, an analytic function assumes every value in $ℂ$ with at most one exception. …

14: 35.1 Special Notation

… ► ►►►

$a,b$	complex variables.
…
$\left|𝐗\right|$	determinant of $𝐗$ (except when $m=1$ where it means either determinant or absolute value, depending on the context).
…

…

15: 28.2 Definitions and Basic Properties

… ►If $q\ne 0$, then for a given value of $\nu$ the corresponding Floquet solution is unique, except for an arbitrary constant factor (Theorem of Ince; see also 28.5(i)). …

16: 15.2 Definitions and Analytical Properties

… ► ►For all values of $c$ … ►Except where indicated otherwise principal branches of $F(a,b;c;z)$ and $𝐅(a,b;c;z)$ are assumed throughout the DLMF. … ►As a multivalued function of $z$, $𝐅(a,b;c;z)$ is analytic everywhere except for possible branch points at $z=0$, $1$, and $\mathrm{\infty }$. The same properties hold for $F(a,b;c;z)$, except that as a function of $c$, $F(a,b;c;z)$ in general has poles at $c=0,-1,-2,\mathrm{\dots }$. …

17: 27.13 Functions

… ►Hilbert (1909) proves the existence of $g\left(k\right)$ for every $k$ but does not determine its corresponding numerical value. The exact value of $g\left(k\right)$ is now known for every $k\le 200,000$. …If $3^k=q{2}^k+r$ with , then equality holds in (27.13.2) provided $r+q\le 2^k$, a condition that is satisfied with at most a finite number of exceptions. ►The existence of $G\left(k\right)$ follows from that of $g\left(k\right)$ because $G\left(k\right)\le g\left(k\right)$, but only the values $G\left(2\right)=4$ and $G\left(4\right)=16$ are known exactly. … ►For values of $k>24$ the analysis of $r_k\left(n\right)$ is considerably more complicated (see Hardy (1940)). …

18: 16.2 Definition and Analytic Properties

… ►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … ►Elsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at $z=0,1$, and $\mathrm{\infty }$. Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … ►In general the series (16.2.1) diverges for all nonzero values of $z$. … ►(However, except where indicated otherwise in the DLMF we assume that when $p>q+1$ at least one of the $a_k$ is a nonpositive integer.) …

19: 4.37 Inverse Hyperbolic Functions

… ►Except where indicated otherwise, it is assumed throughout the DLMF that the inverse hyperbolic functions assume their principal values. …

20: 2.7 Differential Equations

… ►See §2.11(v) for other examples. ►The exceptional case $f_0^2=4g_0$ is handled by Fabry’s transformation: … ►

2.7.25 $${𝒱}_{a_j,x}\left(F\right)=\left|{\int }_{a_j}^x\left|\frac{1}{f^{1/4}(t)}\frac{d^2}{{dt}^2}\left(\frac{1}{f^{1/4}(t)}\right)-\frac{g(t)}{f^{1/2}(t)}\right|dt\right|.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $dx$: differential of $x$ , $\int$: integral, ${𝒱}_{a,b}\left(f\right)$: total variation, $(a_1,a_2)$: interval and $F$: error-control function

Referenced by:

Erratum (V1.1.2) for Equation (2.7.25)

Permalink:

http://dlmf.nist.gov/2.7.E25

Encodings:

pMML, png, TeX

Correction (effective with 1.1.2):

The integrand was corrected so that the absolute value does not include the differential. Also an absolute value was introduced on the right-hand side to ensure a non-negative value.

See also:

Annotations for §2.7(iii), §2.7(iii), §2.7 and Ch.2

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