extended complex plane

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1: 15.11 Riemann’s Differential Equation

… ►A conformal mapping of the extended complex plane onto itself has the form …

2: 14.26 Uniform Asymptotic Expansions

… ►The uniform asymptotic approximations given in §14.15 for $P_{\nu }^{-\mu }\left(x\right)$ and ${𝑸}_{\nu }^{\mu }\left(x\right)$ for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). …

3: 1.9 Calculus of a Complex Variable

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§1.9(iv) Conformal Mapping

►The extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, consists of the points of the complex plane $ℂ$ together with an ideal point $\mathrm{\infty }$ called the point at infinity. …

4: 31.2 Differential Equations

… ►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). …

5: 1.12 Continued Fractions

… ►A sequence $\{C_n\}$ in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be a sequence of convergents of the continued fraction (1.12.3) iff …

6: 1.13 Differential Equations

… ►A domain in the complex plane is simply-connected if it has no “holes”; more precisely, if its complement in the extended plane $ℂ\cup \{\mathrm{\infty }\}$ is connected. …

7: 4.40 Integrals

… ►The results in §§4.40(ii) and 4.40(iv) can be extended to the complex plane by using continuous branches and avoiding singularities. …

8: 4.26 Integrals

… ►The results in §§4.26(ii) and 4.26(iv) can be extended to the complex plane by using continuous branches and avoiding singularities. …

9: 3.4 Differentiation

… ►If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii)) …

10: 14.21 Definitions and Basic Properties

… ►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty })$ to the cut $z$-plane $ℂ\backslash (-\mathrm{\infty },1]$. …