 Solution

ID: past-exam-of-the-algebra-analysis-and-differential-equations-1-course-of-the-university-of-oxford/2013/analysis/6/b/i/solution

f (z) = (z - a)^{m} F (z), g (z) = (z - a)^{n} G (z), F (a) G (a) \neq = 0,

with

F, G

holomorphic near

a

. Shrinking the disk if necessary, continuity ensures

G

never vanishes there, so

F / G

is holomorphic and nonzero at

a

. Consequently

\frac{f ( z )}{g ( z )} = (z - a)^{m - n} \frac{F ( z )}{G ( z )} .

n > m

, the exponent is

- (n - m)

, with a holomorphic nonvanishing factor. Hence

f / g has a pole of order n - m .

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