AbstractA method is described for studying analytic boundary problems for the Laplace equation in a simplyconnected domain D of the plane. The aim is ultimately to obtain solutions in closed form when D is bounded by a simple analytic curve C with Schwarz function G. The basis for the procedure is that a functional F of the incompletely known boundary Cauchy data determines the solution and is analytic in D, and that there exists a further integral relation between these data and the analytic function G. Elimination of the unknown data from the latter relation by using the former leads to an integral on C that relates F to prescribed data and which, in some cases, may be solved for F in D. Some necessary properties of G are derived; in particular, it is shown that C must be a circle if G is analytic in D except for a single, simple pole (Theorem 1), or if a certain integral that appears prominently in this work is analytic throughout D (Theorem 2). The Dirichlet problem is treated in some detail. If C is a circle, a representation for the solution is obtained that is equivalent to one given by P. J Davis, and to the Poisson integral. When G is meromorphic in D and C is not a circle, Theorem 2 implies that the Davis result is not applicable, and the problem is reduced to that of solving Schröder's functional equation. Integral equations for F are obtained when C is an ellipse and G is multivalued in D. Other linear boundary value problems that can be handled are noted. A class of non‐linear Riemann‐Hilbert problems is described and studied briefly. For all these problems, closed solutions have been obtained when C is a circle; it is not clear whether closed solutions for other forms of C can be found without the explicit use of conformal mapping techniques. Finally, some possible generalizations to different equations and more complicated geometries are mentioned.