Abstract
Abstract where an otherwise holomorphic function fails to be holomorphic). Integrating a function round a contour inside which it has one or more singularities will in general give a non-zero result (recall 10.4 and the Cauchy formulae). Understanding, and exploiting, singularities will be the thrust of Chapters 17-22. Our second reason for investigating zeros is more theoretical. Taylor’s theorem implies that holomorphic functions are locally representable by power series, with the coefficients expressible in terms of the derivatives. The consequences of this fact are surprising and far-reaching.
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