Holomorphic function $f: \mathbb{C}^{\times} \to \mathbb{C}$ with essential singularities at $0$ and $\infty$

Question

Is it true that a holomorphic function $f: \mathbb{C}^{\times} \to \mathbb{C}$ can't have an essential singularity at both $0$ and $\infty$ simultaneously? And if so, why is this true?

Assuming $f$ has essential singularities at $0$ and $\infty$, then $g(z) = f(\frac{1}{z})$ also has this property. Then I tried using Laurent series of $f$ and $g$ but didn't get anything out of it. Am I going in the wrong direction?

Answer 1

Any holomorphic function $f: \mathbb{C}^{\times} \to \mathbb{C}$ can be written as a Laurent series $$ f(z) = \sum_{n=-\infty}^{\infty} a_n z^n \, . $$ $f$ has

• an essential singularity at $\infty$ if $a_n \ne 0$ for infinitely many $n > 0$,
• an essential singularity at $0$ if $a_n \ne 0$ for infinitely many $n < 0$.

Every such $f$ can be decomposed into a sum $$ f(z) = g(z) + h(1/z) $$ where $g$ and $h$ are entire functions. Then $f$ has

• an essential singularity at $\infty$ if $g$ is not a polynomial,
• an essential singularity at $0$ if $h$ is not a polynomial.

This gives a complete characterization of all holomorphic functions $f: \mathbb{C}^{\times} \to \mathbb{C}$ with an essential singularity at both $0$ and $\infty$.