## 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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