What is the period

Question

How to find the period of the function \begin{array}{cc} e^{z}\end{array} where "z" is a complex number?? Does this function really have any period??

Answer 1

We say that a function $f(z)$ has period $C$ for some constant $C$ if and only if:

$$ f(z+C) = f(z) $$

for all $z$. In this case the function $e^z$ is defined in the complex plane, so it makes sense to consider complex values for $C$. In particular, since $e^{2\pi i} = 1$:

$$ e^{z+2\pi i} = e^z e^{2\pi i} = e^z $$

So the exponential function is periodic with purely imaginary period $C = 2\pi i$.