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

Depends on

• $\mathbb C=\mathbb R[x]/(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves
• 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
• 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
• 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
• Complex chains, their traces, and cycles Definition
• Function elements and direct analytic continuation Definition
• Homologically simply connected complex domains Definition
• Null-homologous cycles and homologous cycles in an open set 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
• FALSE: a holomorphic function with zero derivative on an arbitrary open set is constant False statement
• FALSE: every continuous complex-valued function on a convex domain has a primitive False statement
• FALSE: every continuous complex-valued function on a domain has a primitive False statement
• FALSE: the local maximum modulus principle needs no connectedness False statement
• A homologically simply connected plane domain has connected spherical complement Lemma
• A plane domain with trivial fundamental group is homologically simply connected Lemma
• Every plane domain has the canonical nested compact exhaustion by distance and radius cutoffs Lemma
• Star-shaped plane domains are homologically simply connected Proposition
• Complex star-shaped and convex domains are the published Euclidean notions under the identification ℂ=ℝ² Remark
• A holomorphic function with zero derivative on a domain is constant Theorem
• A locally uniform limit of injective holomorphic functions is injective or constant Theorem
• A nonvanishing holomorphic function on a homologically simply connected domain has a holomorphic logarithm Theorem
• A plane harmonic function that vanishes on a nonempty open set vanishes everywhere on the domain Theorem
• Equivalent characterisations of a homologically simply connected domain Theorem
• Every holomorphic function on a homologically simply connected domain has a primitive Theorem
• For a continuous function on a complex domain, endpoint independence, zero closed-contour integrals, and existence of a primitive are equivalent Theorem
• Identity theorem for holomorphic functions Theorem
• Liouville's theorem: every bounded entire function is constant Theorem
• Maximum and minimum principles for plane harmonic functions Theorem
• Open mapping theorem for holomorphic functions Theorem

Dependency tree · two levels

18 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)