Invariant under $x \rightarrow 1/x$?

Question

I started thinking on the following problem. I am interested in finding complex functions of a complex variable such that

$\phi(z)=\phi(z^{-1})$

So far, all I could come up with was a family of functions of

$z^{a}+z^{-a}$

where $a$ is a complex parameter. Is there a way to find other functions? In general, a way to say something more on this problem? Thanks! :-)

Answer 1

What about $$\phi(z) = \xi\left(\chi(z)+\chi(z^{-1})\right)$$

with any complex functions $\chi$ and $\xi$

?