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Iii this paper we show that the structure of the Bergman and Szegö kernel functions is especially simple on domains with hyperelliptic double. Each such domain is conformally equivalent to the exterior of a system of slits taken from the real axis, and on such domains the Bergman kernel function and its adjoint are essentially the same, while the Szegö kernel function and its adjoint are elementary and can be written in a closed form involving nothing worse than fourth roots of polynomials. Additionally, a number of applications of these results are obtained. © 1977 American Mathematical Society.