By an exponential polynomial, we shall mean a function Piz) = a0ea»z + • • • + amea"" with arbitrary complex numbers for a's and a's. We shall refer to the a's as the exponents of Piz). We study here functions defined by an equation (1) PnW" + Pn-iw"-1 + • • • + P0 = 0, with every P an exponential polynomial. When the exponents of the P's are all integers, the substitution e' = u converts w into an algebraic function of u. Thus, in a sense, the theory of the equation (1) is a generalization of

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... he theory of algebraic functions. The present investigation arose out of the problem of determining all uniform functions which satisfy an equation (1) . It is a natural conjecture that if w, defined by (1), is uniform, then w = Q/R with Q and R exponential polynomials. One might, further, expect the exponents in Q and R to be linear combinations of the exponents in the P's, with rational coefficients. Both of these conjectures are verified by our work. We have already proved f that if the quotient of two exponential polynomials is an integral function, the quotient is an exponential polynomial. This result is obtained again here, under the weaker hypothesis that the quotient is analytic in a sector of opening greater than -it. The problem of uniformity settled, we are able to discuss the relation between the reducibility properties of (1) and the number of analytic functions which (1) defines. Our result, which generalizes a well known theorem on algebraic functions, is stated in §21. In examining these questions, we were led to study the behavior, for large values of z, of functions determined by (1). It turns out that the Riemann surface for w can be divided into a finite number of sectors, in each of which, after a border of the sector of finite width is removed, w can be represented by a Dirichlet series with complex exponents. The sectors admit of precise description.