 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2011/5/iii/solution

A pole of order

m \geq 1

a

f (z) = \frac{g ( z )}{( z - a ) ^{m}}, g holomorphic near a, g (a) \neq = 0.

Equivalently the lowest nonzero power in its Laurent series is

(z - a)^{- m}

. In particular,

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

tends to a finite nonzero limit.

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