Can all meromorphic functions be described by analytic parametrizations?

Question

Since every meromorphic function $f(z)$ can be written as the quotient of two holomorphic functions, I was wondering whether one can also describe $(z,f(z))$ as parametrized by two analytic functions $(z(t), f(t))$?

As an example, note how $f(z)=1/z$ can be obtained from $z(t)=e^t$ and $f(t)=e^{-t}$.

Answer 1

The example worked fine because there was only one singularity at $z=0$ that could be shifted to infinity by letting $z(t)=e^t$. However, as a result of the "Little" Picard theorem any second singularity could not be omitted, so the answer is:

No, only meromorphic^$\dagger$ functions with one singularity (or none, of course) can be parametrized by analytic functions.

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^$\dagger$ Actually, even functions with one essential singularity or inverse functions can be described this way, e.g.

• $f(t)=t, z(t)=e^t$ for $f(z)=\ln z$
• $f(t)=t,z(t)=t^2$ for $f(z)=\sqrt z$
• $f(t)=\sin e^{-t}, z(t)=e^t$ for $f(z) = \sin 1/z$.