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1: 1.9 Calculus of a Complex Variable

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

Jordan Curve Theorem

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§1.9(iv) Conformal Mapping

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2: Bibliography D

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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.

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External Links:

ISSN 0036-1410, Document, MathReview (H. C. Howard), ZentralBlatt

Referenced by:

§2.8(vi).

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3: 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). …

4: 28.7 Analytic Continuation of Eigenvalues

… ►As functions of $q$, $a_n\left(q\right)$ and $b_n\left(q\right)$ can be continued analytically in the complex $q$-plane. …The number of branch points is infinite, but countable, and there are no finite limit points. In consequence, the functions can be defined uniquely by introducing suitable cuts in the $q$-plane. …The branch points are called the exceptional values, and the other points normal values. … ► …

5: 8.13 Zeros

… ►When $x>x_n^{\ast }$ a pair of conjugate trajectories emanate from the point $a=a_n^{\ast }$ in the complex $a$-plane. …

6: 3.8 Nonlinear Equations

… ►for solving fixed-point problems (3.8.2) cannot always be predicted, especially in the complex plane. …

7: 1.14 Integral Transforms

… ►If the integral converges, then it converges uniformly in any compact domain in the complex $s$-plane not containing any point of the interval $(-\mathrm{\infty },0]$. …

8: 21.7 Riemann Surfaces

… ►Equation (21.7.1) determines a plane algebraic curve in ${ℂ}^2$, which is made compact by adding its points at infinity. …

9: 4.3 Graphics

… ►Figure 4.3.2 illustrates the conformal mapping of the strip onto the whole $w$-plane cut along the negative real axis, where $w={\mathrm{e}}^z$ and $z=\ln w$ (principal value). Corresponding points share the same letters, with bars signifying complex conjugates. Lines parallel to the real axis in the $z$-plane map onto rays in the $w$-plane, and lines parallel to the imaginary axis in the $z$-plane map onto circles centered at the origin in the $w$-plane. In the labeling of corresponding points $r$ is a real parameter that can lie anywhere in the interval $(0,\mathrm{\infty })$. … ►

§4.3(iii) Complex Arguments: Surfaces

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10: 25.12 Polylogarithms

… ►In the complex plane ${\mathrm{Li}}_2\left(z\right)$ has a branch point at $z=1$. …