A complex domain is a nonempty connected open subset of $\mathbb{C}$

Depends on

• The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
• The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
• Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed
• Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets

Used by

• Cauchy's theorem on a convex complex domain Corollary
• Local integrability of measurable conformal structures Corollary
• The modulus of a holomorphic function on a closed polydisc is bounded by its supremum on the distinguished boundary Corollary
• The principal logarithm is the normalised holomorphic branch on the slit plane Corollary
• The ring of holomorphic functions on a complex domain is an integral domain Corollary
• A connected complex domain need not be star-shaped Counterexample
• A connected plane domain that is not homologically simply connected Counterexample
• A holomorphic function on an annulus can have a nonzero closed-contour integral Counterexample
• A nonvanishing holomorphic function on a domain with no holomorphic logarithm Counterexample
• Agreement accumulating only at the boundary does not force a holomorphic identity Counterexample
• Biholomorphic maps between complex domains Definition
• Capacity-polar sets, quasi-everywhere, and subharmonic polar sets Definition
• Complex chains, their traces, and cycles Definition
• Complex sector and bounded analytic semigroup Definition
• Conformal removability of compact sets Definition
• Extremal length and the curve-family modulus of a path family Definition
• Function elements and direct analytic continuation Definition
• Green function with a pole at infinity Definition
• Harmonic measure on a bounded regular plane domain Definition
• Homologically simply connected complex domains Definition
• Measurable Beltrami coefficients and measurable conformal structures Definition
• Null-homologous cycles and homologous cycles in an open set Definition
• Orientation-preserving homeomorphisms and the geometric definition of quasiconformality Definition
• Path integral of a holomorphic differential on a Riemann surface Definition
• The ACL and Sobolev analytic definition of quasiconformality Definition
• The canonical Green kernel of a plane domain Definition
• The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity Definition
• The winding number of a closed contour about a point off its trace Definition
• Weak solutions of the Beltrami equation Definition
• A single point is conformally removable Example
• An irregular puncture does not force the Green kernel to vanish Example
• Composition of two affine quasiconformal maps and the multiplicative dilatation bound Example
• Constant coefficients and their affine solutions Example
• Dolbeault cohomology is independent of hermitian metric Example
• Dolbeault h zero one of the riemann sphere vanishes Example
• Finite and countable planar sets have zero logarithmic capacity Example
• Green kernel of a slit plane via the square-root map Example
• Harmonic measure of the two annulus boundary circles Example
• Infinity-pole Green function recovered from a circular conductor Example
• Nonharmonic exact dbar form Example

…and 64 more results.

Dependency tree · two levels

25 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

• J. Lebl, Guide to Cultivating Complex Analysis, §2.1 (standard reference, not scraped)
• R. Howell and J. Mathews, Complex Analysis, §3.1 (standard reference, not scraped)