What's the inverse function to complex conjugation?

Question

When I say complex conjugation, I mean to define it as a function itself, for example $ \omega = ^-$. How could I define an inverse function to it?

Answer 1

Complex conjugation is its own inverse : $$\overline{\left(\overline{a+bi}\right)}=\overline{a-bi}=a+bi,$$ for all $a,b\in \Bbb R$.

You can also explain this geometrically : in the complex plane, conjugation is the orthogonal symmetry with respect to the real axis, so applying it twice gives you back what you started with.

By the way, a function that is its own inverse is called an involution.