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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 $V$. …
3: 2.1 Definitions and Elementary Properties
If the set $\mathbf{X}$ in §2.1(iii) is a closed sector $\alpha\leq\operatorname{ph}x\leq\beta$, then by definition the asymptotic property (2.1.13) holds uniformly with respect to $\operatorname{ph}x\in[\alpha,\beta]$ as $|x|\to\infty$. …
4: 4.13 Lambert $W$-Function
$W_{0}\left(z\right)$ is a single-valued analytic function on $\mathbb{C}\setminus(-\infty,-{\mathrm{e}}^{-1}]$, 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 $\mathbb{C}\setminus(-\infty,0]$, 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 $\ifrac{{\partial}^{2}p(\alpha,t)}{{\partial t}^{2}}$ 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 $\ifrac{\partial p(\alpha,t)}{\partial t}$ vanishes, but both $\ifrac{{\partial}^{2}p(\alpha,t)}{{\partial t}^{2}}$ 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). …
 $\mathbb{C}$ complex plane (excluding infinity). … $f(z)$ is continuous at all points of a simple closed contour $C$ in $\mathbb{C}$. … closed interval in $\mathbb{R}$, or closed straight-line segment joining $a$ and $b$ in $\mathbb{C}$.
 $(a,b]$ or $[a,b)$ half-closed intervals. … least limit point. …
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 $(\tfrac{1}{2}z)^{\nu}$, is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
10.25.3 $K_{\nu}\left(z\right)\sim\sqrt{\pi/(2z)}e^{-z},$
It has a branch point at $z=0$ for all $\nu\in\mathbb{C}$. The principal branch corresponds to the principal value of the square root in (10.25.3), is analytic in $\mathbb{C}\setminus(-\infty,0]$, and two-valued and discontinuous on the cut $\operatorname{ph}z=\pm\pi$. …
8: 1.10 Functions of a Complex Variable
If $D=\mathbb{C}\setminus(-\infty,0]$ 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 $-\pi<\theta<\pi$ in both cases. Similarly if $D=\mathbb{C}\setminus[0,\infty)$, 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 $0<\theta<2\pi$ 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: …