# extended complex plane

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## 1—10 of 28 matching pages

##### 1: 15.11 Riemann’s Differential Equation

##### 2: 14.26 Uniform Asymptotic Expansions

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►The uniform asymptotic approximations given in §14.15 for ${P}_{\nu}^{-\mu}\left(x\right)$ and ${\bm{Q}}_{\nu}^{\mu}\left(x\right)$ 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).
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##### 3: 1.9 Calculus of a Complex Variable

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###### §1.9(iv) Conformal Mapping

►The*extended complex plane*, $\u2102\cup \{\mathrm{\infty}\}$, consists of the points of the complex plane $\u2102$ together with an ideal point $\mathrm{\infty}$ called the*point at infinity*. …##### 4: 31.2 Differential Equations

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►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $\u2102\cup \{\mathrm{\infty}\}$, can be transformed into (31.2.1).
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##### 5: 1.12 Continued Fractions

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►A sequence $\{{C}_{n}\}$ in the extended complex plane, $\u2102\cup \{\mathrm{\infty}\}$, can be a sequence of convergents of the continued fraction (1.12.3) iff
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##### 6: 1.13 Differential Equations

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►A domain in the complex plane is

*simply-connected*if it has no “holes”; more precisely, if its complement in the extended plane $\u2102\cup \{\mathrm{\infty}\}$ is connected. …##### 7: 4.40 Integrals

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►The results in §§4.40(ii) and 4.40(iv) can be extended to the complex plane by using continuous branches and avoiding singularities.
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##### 8: 4.26 Integrals

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►The results in §§4.26(ii) and 4.26(iv) can be extended to the complex plane by using continuous branches and avoiding singularities.
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##### 9: 3.4 Differentiation

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►If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))
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##### 10: 14.21 Definitions and Basic Properties

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►Many of the properties stated in preceding sections extend immediately from the $x$-interval $(1,\mathrm{\infty})$ to the cut $z$-plane
$\u2102\backslash (-\mathrm{\infty},1]$.
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