Complex zeros of trigonometric polynomials with standard normal random coefficients

In this paper, we obtain an exact formula for the average density of the di stribution of complex zeros of a random trigonometric polynomial eta (0) eta (1) cos theta + eta (2) cos2 theta +(...) + eta (n) cos n theta in (0, 2 pi), where the coefficients eta (J) = a(J) + iotab(J), and {a(J)}(J=1)(n) and {b(J)}(J=1)(n) are sequences of independent normally distributed rando m variables with mean 0 and variance 1. We also provide the limiting behavi our of the zeros density function as n tends to infinity. The corresponding results for the case of random algebraic polynomials are known. (C) 2001 A cademic Press.