This paper describes a method to calculate the potential flow in domains in the complex plane exterior to a finite number of closed curves using conformal mapping. A series method is used to compute the potential flow over multiply connected circle domains. The flow is then mapped from the circle domain to the target physical domain by a method which approximates the Laurent series of the conformal map. The circulations around each boundary can be specified. For flow over multi-element airfoils, the circulations are computed to satisfy the Kutta condition at the trailing edges. The linear systems which are solved on the circle domain for both the potential flow and the conformal maps are of the form identity plus a low-rank matrix, allowing for the efficient use of conjugate-gradient-like methods.