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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. …

2: 1.9 Calculus of a Complex Variable

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Point Sets in $ℂ$

… ►An open set in $ℂ$ is one in which each point has a neighborhood that is contained in the set. … ►A domain $D$, say, is an open set in $ℂ$ that is connected, that is, any two points can be joined by a polygonal arc (a finite chain of straight-line segments) lying in the set. … ►Conversely, if at a given point $(x,y)$ the partial derivatives $\partial u/\partial x$, $\partial u/\partial y$, $\partial v/\partial x$, and $\partial v/\partial y$ exist, are continuous, and satisfy (1.9.25), then $f(z)$ is differentiable at $z=x+\mathrm{i}y$. …

3: 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. …

4: 1.5 Calculus of Two or More Variables

… ►where $D$ is the image of $D^{\ast }$ under a mapping $(u,v)\to (x(u,v),y(u,v))$ which is one-to-one except perhaps for a set of points of area zero. …

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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$(a,b)$	open interval in $ℝ$, or open 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.
$[a_{j,k}]$ or $[a_{jk}]$	matrix with $(j,k)$th element $a_{j,k}$ or $a_{jk}$.
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7: 15.6 Integral Representations

… ►In (15.6.2) the point $1/z$ lies outside the integration contour, $t^{b-1}$ and ${(t-1)}^{c-b-1}$ assume their principal values where the contour cuts the interval $(1,\mathrm{\infty })$, and ${(1-zt)}^a=1$ at $t=0$. …

8: 28.33 Physical Applications

… ►As $\omega$ runs from $0$ to $+\mathrm{\infty }$, with $b$ and $f$ fixed, the point $(q,a)$ moves from $\mathrm{\infty }$ to $0$ along the ray $ℒ$ given by the part of the line $a=(2b/f)q$ that lies in the first quadrant of the $(q,a)$-plane. …

9: 1.16 Distributions

… ►Let $\varphi$ be a function defined on an open interval $I=(a,b)$, which can be infinite. The closure of the set of points where $\varphi \ne 0$ is called the support of $\varphi$. If the support of $\varphi$ is a compact set (§1.9(vii)), then $\varphi$ is called a function of compact support. … ►The set of tempered distributions is denoted by ${𝒯}^{\ast }$. … ►Here $𝜶$ ranges over a finite set of multi-indices, $P(𝐱)$ is a multivariate polynomial, and $P(𝐃)$ is a partial differential operator. …

10: 1.4 Calculus of One Variable

… ►If $f(x)$ is continuous at each point $c\in (a,b)$, then $f(x)$ is continuous on the interval $(a,b)$ and we write $f\in C(a,b)$. … ►where the supremum is over all sets of points in the closure of $(a,b)$, that is, $(a,b)$ with $a,b$ added when they are finite. …