 Solution

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

With the same factors, if

n \leq m

, then

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

has no negative power and extends holomorphically across

a

. Thus its apparent singularity is a removable singularity, with extension value

(\frac{f}{g}) (a) = {F (a) / G (a), 0, m = n, m > n .

For

m > n

the extended quotient has a zero of order

m - n

; for

m = n

the extension is nonzero.

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