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A uniformly convergent sequence of continuous integrands on a fixed contour permits passage of the limit through the complex line integral
Statement
Let be a fixed rectifiable contour. If continuous functions on its trace converge uniformly to a continuous , then
Facts & Assumptions
Given: A rectifiable contour and uniformly convergent continuous functions on its trace.
Continuous integrands have complex line integrals along every rectifiable path (Continuous integrands have complex and absolute line integrals along every rectifiable path).
If on the trace, then (ML estimate: a contour integral is bounded by a supremum bound times path length).
Proof
If , [L2] gives for every .
If , given choose such that on the trace for .
By [L1] all integrals exist, and [L2] applied to gives for .
The two length cases are exhaustive and prove convergence without ever dividing by zero.
Depends on
Used by
- Spectrum can shrink in a larger Banach algebra Counterexample
- Character space of the disc algebra Example
- Riesz projection for a matrix with separated spectrum Example
- The unit-circle integral of exp(z)/z is 2 pi i by uniform termwise integration Example
- A continuous separately holomorphic function is the sum of an absolutely convergent power series with Cauchy-integral coefficients on every smaller polydisc Theorem
- A holomorphic function equals its Taylor series throughout the largest centred disc in its domain Theorem
- Cauchy's integral formula on a circle compactly contained in a disc of holomorphy Theorem
- Laurent coefficients are given by contour integrals and are unique Theorem
- Laurent expansion on an annulus Theorem
- Locally uniform limits of holomorphic functions are holomorphic and their derivatives converge locally uniformly Theorem
Dependency tree · two levels
11 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. 4 (standard reference, not scraped)