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where f, c, u are holomorphic functions of which f, c are known and u is unknown, Cauchy's theorem states the existence of a unique solution. The initial condition (1.2) effectively assigns the coefficients of the terms which involve y alone. Here is proved an existence theorem in which the initial condition assigns one of the (n+1) coefficients of each order n. Corresponding to each non-negative integer n arbitrarily choose a pair of non-negative integers (j., kn) satisfying jn+kn=n. Select an infinite sequence of complex numbers cn subject only to the conditions (i) thatf be holomorphic and f, $0 about x=y=0, u=co, uv=c, where c equals cl or satisfies f(0, 0, co, c) = cl according as ji or ki is 0; (ii) that

A modified initial condition for Cauchy’s existence theorem