Sequences and Series of Analytic Functions, the Residue Theorem
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It is known from real analysis that pointwise convergence of a sequence of functions shows certain pathologies. For instance, the pointwise limit of a sequence of continuous functions is not necessarily continuous, and in general we cannot exchange limit processes, et cetera. Therefore we are led to introduce the notion of uniform convergence, which has better stability properties. For example, the limit of a uniformly convergent sequence of continuous functions is continuous. Another basic stability theorem holds for the (proper) integral:
A uniformly convergent sequence of integrable functions converges to an integrable function. The limit and integration can be exchanged.
However, differentiability in real analysis is not stable with respect to uniform convergence.
The corresponding stability theorems are more complicated and require additional conditions on the sequence of derivatives. In function theory one introduces the concept of uniform convergence by analogy with real analysis. The stability of continuity and of the integral along curves can be obtained completely analogously to the real case, and in fact can be reduced to that case.
KeywordsAnalytic Function Power Series Meromorphic Function Power Series Expansion Laurent Series
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