Summary. In this paper we study the stability and convergence properties of Bergman kernel methods, for the numerical conformal mapping of simply and doubly-connected domains. In particular, by using certain wellknown results of Carleman, we establish a characterization of the level of instability in the methods, in terms of the geometry of the domain under consideration. We also explain how certain known convergence results can provide some theoretical justification of the observed improvement in accuracy which is achieved by the methods, when the basis set used contains functions that reflect the main singular behaviour of the conformal map.

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L e t f b e t h e f u n c t i o n w h i c h m a p s c o n f o r m a l l y a g i v e n d o u b l yconnected domain onto a circular annulus, and let Ω H(z) = f '(z) / f(z)-1/z .