In this paper we consider the space $A_p$ of analytic functions which are $p$-power integrable in a region with an angle. We find a set of numbers $p$ and $q$ ($1/p+1/q=1$) (which depend on the magnitude of the angle) for which the spaces $A_p$ and $A_q$ are mutually conjugate. In each of these spaces we introduce the orthonormal system $$ e_n=\sqrt{(n+1)/\pi}\varphi'\varphi^n,\quad n=0,1,\dots $$ where $\varphi$ is the conformal mapping of the region onto the unit disc. We prove it is dense and determine when it will be a basis.