# 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 $2π × 6 = 12π$.

The answer is accepted.

Join Matchmaticians Affiliate Marketing
Program to earn up to 50% commission on every question your affiliated users ask or answer.

- answered
- 510 views
- $10.00

### Related Questions

- Double Integrals
- Sinusodial graph help (electrical)
- Fixed points of analytic complex functions on unit disk $\mathbb{D}$
- Is $\int_0^1 \frac{\cos x}{x \sin^2 x}dx$ divergent or convergent?
- Why does $ \sum\limits_{n=1}^{\infty } 2^{2n} \times \frac{(n!)^2}{n(2n+1)(2n)!} =2 $ ?
- Evaluate $\int \sqrt{\tan x} dx$
- Let $f\in C (\mathbb{R})$ and $f_n=\frac{1}{n}\sum\limits_{k=0}^{n-1} f(x+\frac{k}{n})$. Prove that $f_n$ converges uniformly on every finite interval.
- real analysis