Cauchy-Riemann equations: Meromorphic Function

Question

A meromorphic function is a function that is holomorphic on all domain except for a set of isolated points. I know that a holomorphic function, by definition, satisfies the Cauchy-Riemann equations but I can't understand if the meromorphic function satisfies the Cauchy-Riemann equations.

Can we restrict meromorphic function to the holomorphic part so it satisfies the Cauchy-Riemann equations? Or due to the set of isolated points the meromorphic function doesn't satisfies the Cauchy-Riemann equations?

Thank you very much for your help!

Answer 1

Off the poles the meromorphic function is holomorphic, so satisfies the C-R equations.

Answer 2

Just to add to Chris' answer, remember in complex analysis everything is done in open sets. So for a given meromorphic function, pick a point $z_0$ where it is actually defined, then since the poles are all isolated you can find an open disk about $z_0$ of non-zero radius such that the function is defined throughout it. By definition, the your function is holomorphic on the disk and therefore satisfies the CR-equations throughout it.