If $f$ is has period $\omega$ and a pole at $z_0$, prove that $f$ has a pole at $z_0+\omega$.

Question

How do I show if $f$ is a meromorphic function with period $\omega$ and a pole at $z_0$, then $z_0+\omega$ is also a pole of $f$ with the same multiplicity and residue?

Answer 1

Assuming $\epsilon>0$ is small enough to contain only one pole of $f$, we follow Cauchy... $$\oint_{|z_0+\omega-z|=\epsilon} \frac{f(z)}{z-(z_0 + \omega)}\mathrm{d}z = \oint_{|z_0-y|=\epsilon} \frac{f(y+\omega)}{y-z_0}\mathrm{d}y = \oint_{|z_0-y|=\epsilon} \frac{f(y)}{y-z_0}\mathrm{d}y$$ where we have used the substitution $y = z-\omega$. This last integral should look pretty familiar.