# When is composition of meromorphic functions meromorphic

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 $$\lim_{z\to z_0} f(g(z))$$ 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 $$\Bbb C$$.

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 $$f = e^z$$, $$g = \frac1z$$, ...