Maximum value of tanh Z
$$ f(z) = \tanh (z) = \dfrac{e^z - e^{-z}}{e^z + e^{-z}} $$
Find the point $z$ with $|z| \leq1$ where $|f(z)|$ attain its maximum.
I figured out that the maximum is probably at the edge (concluded it from cauchy integral formula ) but I am not sure what is my next step (In section a,b of the question I found the taylor expansion around $z=0$ and the radius of convergence)
Can someone help ?
$\endgroup$1 Answer
$\begingroup$If $|z|=1,$ you can write $\tanh(z)=\frac {e^{e^{i\theta}}-e^{e^{-i\theta}}}{e^{e^{i\theta}}+e^{e^{-i\theta}}}$, then in theory can take absolute value and the derivative with respect to $\theta$. This Alpha plot indicates the maxima are at $z=\pm i$, with $|\tanh z| = \tan 1 \approx 1.5574$
$\endgroup$ 5
