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The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity

Definition

Let $U_{0} : = \hat{C} ∖ {\infty}, U_{\infty} : = \hat{C} ∖ {0} .$ Define charts $ϕ_{0} : U_{0} \to C, ϕ_{0} (z) = z,$ and $ϕ_{\infty} : U_{\infty} \to C, ϕ_{\infty} (z) = {\begin{cases} 1 / z, & z \in C^{\times}, \\ 0, & z = \infty . \end{cases}$ On the overlap $U_{0} \cap U_{\infty} = C^{\times}$ the transition maps are $ϕ_{\infty} \circ ϕ_{0}^{- 1} (w) = 1 / w, ϕ_{0} \circ ϕ_{\infty}^{- 1} (w) = 1 / w,$ which are holomorphic on $C^{\times}$ . These are the standard holomorphic charts of the Riemann sphere.

If $V \subseteq \hat{C}$ is open and $f : V \to C$ , then $f$ is holomorphic at $\infty$ when $\infty \in V$ and the chart expression $F (w) : = f (ϕ_{\infty}^{- 1} (w)) = {\begin{cases} f (1 / w), & w \neq 0, \\ f (\infty), & w = 0 \end{cases}$ is holomorphic at $0$ . A scalar-valued function defined on a punctured neighbourhood of $\infty$ has a pole at $\infty$ when the punctured chart expression $w \mapsto f (1 / w)$ has a pole at $0$ in the sense of Isolated singularities: removable, poles, and essential singularities.

Depends on

Used by

Meromorphic functions on the Riemann sphere Definition
Every Möbius transformation is a biholomorphism of the Riemann sphere Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lars V. Ahlfors, Complex Analysis, 3rd ed., Ch. 3 §§2.2-3.5 (standard reference, not scraped)
Matthias Weber, Complex Analysis, Ch. 1 §§1.3-1.4 (standard reference, not scraped)
Elias M. Stein and Rami Shakarchi, Complex Analysis, Ch. 8 §§1-2 (standard reference, not scraped)