 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2004/analysis/5/b/solution

A pole of order

m \geq 1

means that

(z - a)^{m} f (z)

extends holomorphically to

a

and has a nonzero value there. Equivalently, the lowest nonzero power in the Laurent expansion is

(z - a)^{- m}

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