Ggtkuk

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?

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 $lim_zto z_0 f(g(z))$ 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 $Bbb C$.

The composition of holomorphic functions is holomorphic. As a special case, a holomorphic function followed by a meromorphic function is meromorphic.

I say 'special case' because we can think of a meromorphic function as a holomorphic function $mathbb C to P^1(mathbb C)$.

The converse is often false. In fact, when:

• 
  $f$ is entire and there is a value which it reaches infinitely often


• 
  $g$ is meromorphic with at least one pole

then $f circ g$ is not meromorphic unless $f$ is constant. Indeed, $f circ g$ reaches a value infinitely often in any neighborhood of the pole of $g$, which can only be if $f circ g$ is constant.

Examples: $f = sin z$, $g = frac1z$, or $f = e^z$, $g = frac1z$, ...