I came across the following question: If a complex function $f(z)$ is real on the x-axis and Laurent expansion about origin is $f(z)=\sum_{n=-\infty}^{\infty} a_nz^n$ has $a_n=0$ for $n<-N$, then show that the coefficients $a_n$ are real.
I attempted the solution: I considered $z^N f(z)=\sum_{n=0}^{\infty} a_{(-N+n)}z^n$. This has no poles about the origin, hence by residue theorem, the integral over any closed contour of $z^N f(z)$ has to be zero. Now, by Morera's theorem $z^N f(z)$ is analytic and hence has the unique Laurent expansion $z^N f(z)=\sum_{n=0}^{\infty} a_{(-N+n)}z^n$. Since $f(z)$ is real when $z$ is real, we can conclude $a_n$ is real.
Is this proof correct?
Thanks in advance.