# Rouche’s Theorem applied to the complex valued function $f(z) = z^6 + \cos z$

## Answer

\[|g(z)| = |z| ^6 = 26 = 64\] and

\[|f(z) − g(z)| = | \cos(z)|= \frac{|e ^{iz} + e^{ −iz}|}{ 2}\leq \frac{|e ^{iz}| + |e^{ −iz}|}{ 2} =e^{ |z|} = e ^2 < 64.\]

Thus, Rouche’s Theorem applies and $f(z)$ has the same number of zeros inside the circle as $z^6$ , which is $6$. Thus, the change in argument is $2π × 6 = 12π$.

Erdos

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