Rouche's Theorem

We will now look at a very important and relatively simple theorem in complex analysis known as Rouche's theorem. This theorem gives us a method to determine the number of roots of a function (counting multiplicities) in a region under certain conditions.

Theorem 1 (Rouche's Theorem): Let $A \subseteq \mathbb{C}$ be open and let $\gamma$ be any positively oriented, simple, closed, piecewise smooth curve in $A$ that is homotopic to a point in $A$. If: 1) $f$ and $g$ are analytic on $A$. 2) $f$ and $g$ have no roots of $\gamma$. 3) $\mid f(z) - g(z) \mid < \mid f(z) \mid$ for all $z \in \gamma$. Then $f$ and $g$ have the same number of zeroes (counting multiplicities) inside of $\gamma$.

For example, consider the function $g(z) = e^z + 5z^3 + 1$. We claim that this functions has $3$ roots (counting multiplicities) inside the unit circle.

To show this, consider the function $f(z) = 5z^3$. Note that $f$ has $3$ roots (counting multiplicities) inside the unit circle. Note that $f$ and $g$ are analytic on $\mathbb{C}$, $f$ and $g$ have no roots on the unit circle, and on the unit circle:

(1)

\begin{align} \quad \mid f(z) - g(z) \mid = \mid 5z^3 - e^z + 5z^3 + 1 \mid = \mid -e^z + 1 \mid \leq \mid e^z \mid + 1 \leq 2 < 5 = \mid 5z^3 \mid = \mid f(z) \mid \end{align}

Therefore, by Rouche's theorem we must have that $f$ and $g$ contain the same number of roots (counting multiplicities) in $D(0, 1)$, i.e., $g$ contains $3$ roots inside of $D(0, 1)$.