# extended complex plane

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## 1—10 of 28 matching pages

##### 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 $P^{-\mu}_{\nu}\left(x\right)$ and $\boldsymbol{Q}^{\mu}_{\nu}\left(x\right)$ for $1 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
###### §1.9(iv) Conformal Mapping
The extended complex plane, $\mathbb{C}\,\cup\,\{\infty\}$, consists of the points of the complex plane $\mathbb{C}$ together with an ideal point $\infty$ 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, $\mathbb{C}\cup\{\infty\}$, can be transformed into (31.2.1). …
##### 5: 1.12 Continued Fractions
A sequence $\{C_{n}\}$ in the extended complex plane, $\mathbb{C}\cup\{\infty\}$, 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 $\mathbb{C}\cup\{\infty\}$ 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 $(1,\infty)$ to the cut $z$-plane $\mathbb{C}\backslash(-\infty,1]$. …