  Let $\{\eta_{j}\}_{j = 0}^{N}$ be a sequence of independent, identically
distributed random complex Gaussian variables, and let $\{f_{j} (z)\}_{j =
0}^{N}$ be a sequence of given analytic functions that are real-valued on the
real number line. We prove an exact formula for the expected density of the
distribution of complex zeros of the random equation $\sum_{j = 0}^{N} \eta_{j}
f_{j} (z) = \mathbf{K}$, where $\mathbf{K} \in \mathds{C}$. The method of proof
employs a formula for the expected absolute value of quadratic forms of
Gaussian random variables. We then obtain the limiting behaviour of the density
function as $N$ tends to infinity and provide numerical computations for the
density function and empirical distributions for random sums with certain
functions $f_{j} (z)$. Finally, we study the case when the $f_{j} (z)$ are
polynomials orthogonal on the real line and the unit circle.
