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1: 1.6 Vectors and Vector-Valued Functions

… ►Note: The terminology open and closed sets and boundary points in the $(x,y)$ plane that is used in this subsection and §1.6(v) is analogous to that introduced for the complex plane in §1.9(ii). … ►and $S$ be the closed and bounded point set in the $(x,y)$ plane having a simple closed curve $C$ as boundary. … ►Suppose $S$ is a piecewise smooth surface which forms the complete boundary of a bounded closed point set $V$, and $S$ is oriented by its normal being outwards from $V$. …

2: 1.9 Calculus of a Complex Variable

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3: 2.1 Definitions and Elementary Properties

… ►If the set $𝐗$ in §2.1(iii) is a closed sector $\alpha \le \mathrm{ph}x\le \beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\mathrm{ph}x\in [\alpha ,\beta ]$ as $|x|\to \mathrm{\infty }$. …

4: 4.13 Lambert $W$-Function

… ► $W_0\left(z\right)$ is a single-valued analytic function on $ℂ\setminus (-\mathrm{\infty },-{\mathrm{e}}^{-1}]$, real-valued when $z>-{\mathrm{e}}^{-1}$, and has a square root branch point at $z=-{\mathrm{e}}^{-1}$. …The other branches $W_k\left(z\right)$ are single-valued analytic functions on $ℂ\setminus (-\mathrm{\infty },0]$, have a logarithmic branch point at $z=0$, and, in the case $k=\pm 1$, have a square root branch point at $z=-{\mathrm{e}}^{-1}\mp 0\mathrm{i}$ respectively. …

5: 2.3 Integrals of a Real Variable

… ►Assume also that ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are continuous in $\alpha$ and $t$, and for each $\alpha$ the minimum value of $p(\alpha ,t)$ in $[0,k)$ is at $t=\alpha$, at which point $\partial p(\alpha ,t)/\partial t$ vanishes, but both ${\partial }^{2}p(\alpha ,t)/{\partial t}^2$ and $q(\alpha ,t)$ are nonzero. …

6: Mathematical Introduction

… ►These include, for example, multivalued functions of complex variables, for which new definitions of branch points and principal values are supplied (§§1.10(vi), 4.2(i)); the Dirac delta (or delta function), which is introduced in a more readily comprehensible way for mathematicians (§1.17); numerically satisfactory solutions of differential and difference equations (§§2.7(iv), 2.9(i)); and numerical analysis for complex variables (Chapter 3). … ► ►►►►

$ℂ$	complex plane (excluding infinity).
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${f(z)|}_C=0$	$f(z)$ is continuous at all points of a simple closed contour $C$ in $ℂ$.
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$[a,b]$	closed interval in $ℝ$, or closed straight-line segment joining $a$ and $b$ in $ℂ$.

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$(a,b]$ or $[a,b)$	half-closed intervals.
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$lim\; inf$	least limit point.
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7: 10.25 Definitions

… ►In particular, the principal branch of $I_{\nu }\left(z\right)$ is defined in a similar way: it corresponds to the principal value of ${(\frac{1}{2}z)}^{\nu }$, is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. … ►

10.25.3 $$K_{\nu }\left(z\right)\sim \sqrt{\pi /(2z)}{\mathrm{e}}^{-z},$$

ⓘ

Defines:

$K_{\nu }\left(z\right)$: modified Bessel function of the second kind

Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.25(ii), §10.25(iii), §10.30(ii)

Permalink:

http://dlmf.nist.gov/10.25.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.25(ii), §10.25 and Ch.10

… ►It has a branch point at $z=0$ for all $\nu \in ℂ$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $ℂ\setminus (-\mathrm{\infty },0]$, and two-valued and discontinuous on the cut $\mathrm{ph}z=\pm \pi$. …

8: 1.10 Functions of a Complex Variable

… ►If $D=ℂ\setminus (-\mathrm{\infty },0]$ and $z=r{\mathrm{e}}^{\mathrm{i}\theta }$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. Similarly if $D=ℂ\setminus [0,\mathrm{\infty })$, then one branch is $\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, the other branch is $-\sqrt{r}{\mathrm{e}}^{\mathrm{i}\theta /2}$, with in both cases. … ► … ►Alternatively, take $z_0$ to be any point in $D$ and set $F(z_0)={\mathrm{e}}^{\alpha \ln \left(1-z_0\right)}{\mathrm{e}}^{\beta \ln \left(1+z_0\right)}$ where the logarithms assume their principal values. … ►Let $D$ be a domain and $[a,b]$ be a closed finite segment of the real axis. …

9: 1.8 Fourier Series

… ►If a function $f(x)\in C^2[0,2\pi ]$ is periodic, with period $2\pi$, then the series obtained by differentiating the Fourier series for $f(x)$ term by term converges at every point to $f^{\prime }(x)$. …

10: 3.1 Arithmetics and Error Measures

… ►A nonzero normalized binary floating-point machine number $x$ is represented as … … ► … ►Let $G$ be the set of closed intervals $\{[a,b]\}$. The elementary arithmetical operations on intervals are defined as follows: …