2012/07/05 by Oswaldo Rio Branco de Oliveira, de Oliveira, Oswaldo Rio Branco
Mathematics · #30B10 #40-01 #40B05 #40C15 #97I80 #Advanced Mathematical Identities #Advanced Mathematical Theories #Analytic and geometric function theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #math.CV #msc:30B10 #msc:40-01 #msc:40B05 #msc:40C15 #msc:97I80
paper · pdf · doi:10.48550/arxiv.1207.1472
18 pages, corrected typos, notations removed in sections 3 and 4, a lemma introduced and a result improved both in section 8, a name added in Acknowledgments
openalex publication_date 2012/07/05 · arxiv created 2012/07/27 · arxiv updated 2012/07/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This text provides very easy and short proofs of some basic properties of complex power series (addition, subtraction, multiplication, division, rearrangement, composition, differentiation, uniqueness, Taylor's series, Principle of Identity, Principle of Isolated Zeros, and Binomial Series). This is done by simplifying the usual presentation of unordered sums of a (countable) family of complex numbers. All the proofs avoid formal power series, double series, iterated series, partial series, asymptotic arguments, complex integration theory, and uniform continuity. The use of function continuity as well as epsilons and deltas is kept to a minimum.