of point sets in complex plane

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1: 1.9 Calculus of a Complex Variable

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Point Sets in $ℂ$

… ►Also, the union of $S$ and its limit points is the closure of $S$. … ► … ► … ►

Jordan Curve Theorem

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2: 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). …

3: 21.7 Riemann Surfaces

… ►Belokolos et al. (1994, §2.1)), they are obtainable from plane algebraic curves (Springer (1957), or Riemann (1851)). Consider the set of points in ${ℂ}^2$ that satisfy the equation …Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. … ►Riemann theta functions originating from Riemann surfaces are special in the sense that a general $g$-dimensional Riemann theta function depends on $g(g+1)/2$ complex parameters. … ►Denote the set of all branch points by $B=\{P_1,P_2,\mathrm{\dots },P_{2g+1},P_{\mathrm{\infty }}\}$. …

4: 36.5 Stokes Sets

… ►where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\mathrm{\Re }\mathrm{\Phi }=\mathrm{constant}$) in complex $t$ or $(s,t)$ space. … ►The Stokes set consists of the rays $\mathrm{ph}x=\pm 2\pi /3$ in the complex $x$-plane. … ►In Figures 36.5.1–36.5.6 the plane is divided into regions by the dashed curves (Stokes sets) and the continuous curves (bifurcation sets). Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …

5: Mathematical Introduction

… ►(These chapters can also serve as background material for university graduate courses in complex variables, classical analysis, and numerical analysis.) … ►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). … ►Special functions with a complex variable are depicted as colored 3D surfaces in a similar way to functions of two real variables, but with the vertical height corresponding to the modulus (absolute value) of the function. …However, in many cases the coloring of the surface is chosen instead to indicate the quadrant of the plane to which the phase of the function belongs, thereby achieving a 4D effect. … ►In the DLMF this information is provided in pop-up windows at the subsection level. …

6: 20.2 Definitions and Periodic Properties

… ►For fixed $z$, each of ${\theta }_1\left(z\right|\tau )/\sin z$, ${\theta }_2\left(z\right|\tau )/\cos z$, ${\theta }_3\left(z\right|\tau )$, and ${\theta }_4\left(z\right|\tau )$ is an analytic function of $\tau$ for $\mathrm{\Im }\tau >0$, with a natural boundary $\mathrm{\Im }\tau =0$, and correspondingly, an analytic function of $q$ for with a natural boundary $\left|q\right|=1$. ►The four points $(0,\pi ,\pi +\tau \pi ,\tau \pi )$ are the vertices of the fundamental parallelogram in the $z$-plane; see Figure 20.2.1. The points …are the lattice points. The theta functions are quasi-periodic on the lattice: …

7: 3.8 Nonlinear Equations

… ► … ►These results are also useful in ensuring that no zeros are overlooked when the complex plane is being searched. … ►for solving fixed-point problems (3.8.2) cannot always be predicted, especially in the complex plane. … ►For an arbitrary starting point $z_0\in ℂ$, convergence cannot be predicted, and the boundary of the set of points $z_0$ that generate a sequence converging to a particular zero has a very complicated structure. …

8: 25.12 Polylogarithms

… ►The notation ${\mathrm{Li}}_2\left(z\right)$ was introduced in Lewin (1981) for a function discussed in Euler (1768) and called the dilogarithm in Hill (1828): … ►In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. … ►The cosine series in (25.12.7) has the elementary sum … ►(In the latter case (25.12.11) becomes (25.5.1)). … ►Sometimes the factor $1/\mathrm{\Gamma }\left(s+1\right)$ is omitted. …

9: 3.1 Arithmetics and Error Measures

… ► … ► … ►The current floating point arithmetic standard is IEEE 754-2019 IEEE (2019), a minor technical revision of IEEE 754-2008 IEEE (2008), which was adopted in 2011 by the International Standards Organization as ISO/IEC/IEEE 60559. … ►where $\mathrm{\ast }\in \{\mathrm{+},\mathrm{-},\mathrm{\cdot },\mathrm{/}\}$, with appropriate roundings of the end points of $I\ast J$ when machine numbers are being used. … ►The last reference includes analogs for arithmetic in the complex plane $ℂ$. …

10: 22.4 Periods, Poles, and Zeros

… ►For each Jacobian function, Table 22.4.1 gives its periods in the $z$-plane in the left column, and the position of one of its poles in the second row. The other poles are at congruent points, which is the set of points obtained by making translations by $2mK+2n\mathrm{i}{K}^{\prime }$, where $m,n\in \mathbb{Z}$. … ►Table 22.4.2 displays the periods and zeros of the functions in the $z$-plane in a similar manner to Table 22.4.1. … ►The other poles and zeros are at the congruent points. … ►The set of points $z=mK+n\mathrm{i}{K}^{\prime }$, $m,n\in \mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime }$, where again $m,n\in \mathbb{Z}$. …