How to solve th(z)=0?

Question

Hyperbolic tangent How to find all the solutions to th(z)=0 ? I do have the solution z=kπi but it does not make any sense to me

Answer 1

LINK $$ \tanh z = \frac{e^{2z}-1}{e^{2z}+1} $$ for complex number $z$.
Therefore $\tanh z = 0$ if and only if $e^{2z}=1$. That means $2z = 2\pi i n$ for some integer $n$, so $z = \pi i n$ for some integer $n$. (Of course $i^2 = -1$ which we use to build the complex numbers.)

The only real number $z$ with $\tanh z = 0$ is $z=0$. All other solutions are purely imaginary.