Find the maximum of a |cos(z)| How do you find the maximum of the complex function $|\cos{z}|$ on $[0,2\pi]\times[0,2\pi]$. I believe I'm to use the maximum modulus principle, since the function is entire. I'm just having problems starting. Any suggestions?
 A: Because cos z is entire, |cos z| has its maximum along the boundary of the square [0,2$\pi$]$\times$[0,2$\pi$], by the Maximum Modulus Principle.
Let cos z = cos($x+iy$), so that we can check its values at the boundaries. Although cos $\theta$ for real values just oscillates between 1 and -1, we are not dealing with real values.
It just so happens that |cos($\tilde{x}+2\pi i$)|=267.746761483748... (a transcendental number) for $\tilde{x}=\frac{n\pi}{2}$ for n=0,1,2,3,4. You can show this using the exponential representation of cos($x+iy$). Thus, the maximum is 267.746761483748...
A: The idea is to apply the same technique used in one of the answers of this problem Maximum of $|\sin(z)|$ as $\{z: |z| \leq 1 \} $ 
Use the identity: $$\vert\cos z\vert^2=\vert \cos (x+iy)\vert^2=\cos^2 x + \sinh ^2 y$$. 
What is left is just to find the maximum of $\cos x$ and the maximum of $\sinh y$ in the boundary. Since $x, y \in[0,2\pi]$, the maximum of $\cos x$ is $1$ when $x=0,\,2\pi$ and the maximum of $\sinh$ is $267.74489404102$ when $y=2\pi$.
Finally by the Maximum Modulus Principle $$\max_{[0,2\pi]\times[0,2\pi]}\vert \cos z\vert=\sqrt{1^2+267.74489404102^2}\approx 267.74676148375$$
