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The discrete Theodorsen equation is the basis of efficient numerical methods for conformal mappings of the unit disk onto Jordan regions that are starlike with respect to the origin. Applying Brouwer's fixed point theorem we show here that there always exists a solution to this equation. Let D denote the unit disk (in the z-plane) and G a region (in the w-plane) with boundary r, where r is a Jordan curve that is starlike with respect to the origin and thus defined by polar coordinates r, p(r), p being a 2rr-periodic positive continuous function. (Note that 'starlike' sometimes is defined slightly more general.) Then there exists a unique mapping function f: D G that is continuous in D, conformal in D, and normalized by f(O) = 0, f'(0) > O. f is already determined by a real 27rperiodic function 0 satisfying f(ecI) p [O()] elO (f). For one of these functions Theodorsen's integral equation holds [2, p. 65]: (1) GQP) = ? + 1 PV[ 2 ( log p[0(a)] cot 2 a do. The existence of f implies that this nonlinear singular integral equation always has at least one strictly monotone continuous solution. One can show [2, Theorem 1.3, p. 66] that there exists only one solution with these properties. Moreover, if r satisfies a so-called e-condition with e < 1, then (1) cannot have additional nonmonotone continuous solutions. Definition. A starlike Jordan curve defined by polar coordinates r,, p(r) satisfies an e-condition if p is absolutely continuous and p 'C) (2) e:= ess sup | p-r0j| <0 For numerical computations Eq. (1) needs to be discretized. The easiest and most useful way to do this [21, [3] leads to what we call the discrete Theodorsen equation: (3) = + K log p(G). Here k, 0,-log p(O) are n-dimensional vectors (n even) with components 'k: = 2k/ci/n, Ok' and log p (0k), k = 1,... , n, respectively, and K is Wittich's matrix: Received July 8, 1976. AMS (MOS) subject classifications (1970)Primary 30A28; Secondary 65R05.

Existence of a solution to the discrete Theodorsen equation for conformal mappings