A Fourier Series Method for Meromorphic and Entire Functions

摘要

If f is a meromorphic function in the complex plane and if c sub k(r,f) is the k-th Fourier coefficient of log (absolute value of f(r(e to the power i theta))), then the behavior of f is reflected in the behavior of the sequence (c sub k(r,f)) and vice versa. Further, the c sub k(r,f) are determined by the distribution of the zeros and the poles of f and the local behavior of f near the origin. Applications are made to the value distribution theory of meromorphic functions of a given rate of growth. Necessary and sufficient conditions are presented that each meromorphic function of a given rate of growth be the quotient of two entire functions of the same rate of growth. A so-called 'Generalized Hadamard Product' is developed. (Author)