Uniformization of simply connected finite type log-Riemann surfaces

Abstract: We consider simply connected log-Riemann surfaces with a finite number of ramification points. We prove that these surfaces are biholomorphic to C with uniformizations given by entire functions of the form F (z) = \int Q(z) e^{P(z)} dz where P, Q are polynomials of degrees equal to the number of infinite and finite order ramification points respectively. Conversely any such entire function defines a simply connected log-Riemann surface with finitely many ramification points.