The Complex Roots of Random Sums

This paper extends earlier work on the distribution in the complex plane of the roots of random polynomials. In this paper, the random polynomials are generalized to random finite sums of given "basis" functions. The basis functions are assumed to be entire functions that are real-valued on the real line. The coefficients are assumed to be independent identically distributed Normal $(0,1)$ random variables. An explicit formula for the density function is given in terms of the set of basis functions. We also consider some practical examples including Fourier series. In some cases, we derive an explicit formula for the limiting density as the number of terms in the sum tends to infinity.