Rouche’s Theorem

Rouche’s Theorem

If the complex-valued functions f and g are holomorphic inside and on some closed contour K, with |g(z)|<|f(z)| on K, then f and f+g have the same number of zeroes inside K, where each zero is counted as many times as its multiplicity.

Example

Consider the polynomial z^5+3z^3+7 in the disk |z|<2. Let g(z)=3z^3+7, f(z)=z^5, then

\begin{aligned}  |3z^3+7|&<3(8)+7\\  &=31\\  &<32\\  &=|z^5|  \end{aligned}
for every |z|=2.
Then f+g has the same number of zeroes as f(z)=z^5 in the disk |z|<2, which is exactly 5 zeroes.

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One Response to Rouche’s Theorem

  1. LispMobile says:

    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…

    Liked by 1 person

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