TITLE: When is composition of meromorphic functions meromorphic
QUESTION [6 upvotes]: When I compose a meromorphic and a holomorphic function, I get a meromorphic function. Are there other cases when a composition of two meromorphic functions is meromorphic? For example, if I compose a holomorphic and a meromorphic function? Or does it hold that the composition is meromorphic in general?

REPLY [13 votes]: Assume that $f$ and $g$ are meromorphic in $\Bbb C$, and that the composition $h = f \circ g$ is also meromorphic in $\Bbb C$. 
If $f$ is not a rational function then it has an essential singularity at $z= \infty$. If in addition, $g$ has a pole at $z_0$, then the Casorati-Weierstraß theorem shows that $\lim_{z\to z_0} f(g(z))$ does not exist in the extended complex plane. So this can not happen.
Therefore $f \circ g$ is meromorphic in $\Bbb C$ if and only if

$f$ is a rational function, or
$g$ is holomorphic in $\Bbb C$.