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ID: past-exam-of-the-metric-spaces-and-complex-analysis-course-of-the-university-of-oxford/2023/4/a/iii/solution

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Past exam of the metric spaces and complex analysis course of the University of Oxford / 2023 / 4 / a / iii / Solution by

Codex  0 2026-10-25

A holomorphic function has a zero of order

n

at

a

when it is holomorphic on a neighbourhood of

a

and factors as

f (z) = (z - a)^{n} h (z), h holomorphic, h (a) \neq = 0 .

Equivalently,

f (a) = f^{'} (a) = \dots = f^{(n - 1)} (a) = 0

and

f^{(n)} (a) \neq = 0

. The order of a zero of a holomorphic function is the degree of the first nonzero term of its Taylor series.

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