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Uniform convergence and the uniformly Cauchy condition for complex-valued functions, with the componentwise dictionary
Definition
Let be a set and let . The sequence converges uniformly to when It is uniformly Cauchy when
Writing and , uniform convergence in complex modulus is equivalent to uniform convergence of both real component sequences. This follows from and . These are the complex analogues of Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions, using the metric of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane and the componentwise convergence clause in The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts.
Depends on
- Pointwise convergence, uniform convergence, and the uniformly Cauchy condition for sequences of real-valued functions
- The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
- The complex plane is complete, and convergence is equivalent to convergence of real and imaginary parts
Used by
- The complex geometric series is not uniformly convergent on its open unit disc Counterexample
- zⁿ tends locally uniformly to zero on the unit disc but not uniformly on the closed disc Counterexample
- Multi-indexed power series in ℂᵐ and their absolute convergence Definition
- The Cauchy kernel expands as an absolutely and uniformly convergent multi-indexed geometric series Lemma
- A sequence of complex-valued functions converges uniformly if and only if it is uniformly Cauchy Theorem
- A uniform limit of continuous complex-valued functions is continuous Theorem
- Locally uniform limits of holomorphic functions are holomorphic, with locally uniform convergence of all derivatives Theorem
- Weierstrass M-test for complex-valued function series Theorem
Dependency tree · two levels
14 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
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)