Consider two functions $f(z)$ and $g(\bar{z})$ where $z=x+iy$ and $\bar{z} = x-iy$. It is given that these functions admit a Laurent expansion around $z=0$ as $$f(z) = \sum_{n=-\infty}^{\infty} f_n z^n \quad g(\bar{z}) = \sum_{n=-\infty}^{\infty}g_n \bar{z}^n$$ It is also given that these functions are equal on the real axis i.e. $$f(z) = g(\bar{z}) \text{ for } z = \bar{z}$$ What can we conclude about the relation between the coefficients $f_n$ and $g_n$? Would $f_n=g_n$ hold $\forall n\in Z$?
In other words, if the Laurent expansion (around $z=0$) of two complex functions is equal on the real axis, can we equate the coefficients?
Let $h(z)= \sum_{n=-\infty}^{\infty}g_n {z}^n$; by hypothesis $h$ is analytic nn the punctured plane and $f=h$ on the real axis, so $f=h$ everywhere, hence $f_n=g_n$