The group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}[x,y]$ is well-known, see, for example, the proof of Dicks or the proof of Mckay and Wang.

What can be said about the group of $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$?

Of course, every $\mathbb{C}$-algebra automorphism of $\mathbb{C}[x,y]$ yields a $\mathbb{C}$-algebra automorphism of $\mathbb{C}(x,y)$, but there are more $\mathbb{C}$-algebra automorphisms of $\mathbb{C}(x,y)$, for example, $x \mapsto x^{-1}, y \mapsto y$.

Can one find all of them and characterize that group?