Does Euler formula work for complex $z$?

Question

We know Euler formula:

$$ e^{ix}=\cos(x)+i\sin(x). $$

Does this formula work if we replace real number $x$ with complex number $z$?

Answer 1

Yes, it does. In fact its validity for real $x$ implies its validity for complex $x$ also because the real line has limit points in the complex plane. [If two analytic functions are equal on a set with limit points then they are equal everywhere].

Answer 2

Just let $z=a+i b$

$$e^{i z}=e^{i (a+i b)}=e^{ia-b}=e^{-b}e^{ia}=e^{-b}\big(\cos(a)+i \sin(a)\big)$$