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The sum of a complex power series is analytic throughout its open disc of convergence
Statement
The sum of a complex power series is analytic on its open disc of convergence.
Facts & Assumptions
Given: A complex power-series sum on its open disc .
At every interior point , the sum re-expands as a convergent power series on a positive-radius disc about (A complex power-series sum re-expands about every interior point, at least to the distance from that point to the original boundary).
Analyticity means local representation by a convergent complex power series (Complex analytic functions as locally representable by convergent power series).
Proof
Let . Its distance to the original boundary is positive, and [L1] supplies a power-series representation of on a disc about .
This is exactly analyticity at by [L2]. Since was arbitrary, is analytic on , including the entire-radius case.
Depends on
Used by
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- Zero free entire function of exponential type is an exponential Lemma
- The associated variety is a closed conical coadjoint-invariant cone Proposition
- Gleason Kahane Zelazko Theorem
- Green and harmonic-measure representation with the 2π sign 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
- L. Ahlfors, Complex Analysis, 3rd ed., Ch. 2 (standard reference, not scraped)