Rouche’s Theorem
If the complex-valued functions and
are holomorphic inside and on some closed contour
, with
on
, then
and
have the same number of zeroes inside
, where each zero is counted as many times as its multiplicity.
Example
Consider the polynomial in the disk
. Let
,
, then
for every .
Then has the same number of zeroes as
in the disk
, which is exactly 5 zeroes.
Intuitively since f > g, so ( f+g) doesn’t change the numbers of zero of f even it adds on g. The bigger f just covers the smaller g…
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