  The zeros of the random Laurent series $1/\mu - \sum_{j=1}^\infty c_j/z^j$,
where each $c_j$ is an independent standard complex Gaussian, is known to
correspond to the scaled eigenvalues of a particular additive rank 1
perturbation of a standard complex Gaussian matrix. For the corresponding
random Maclaurin series obtained by the replacement $z \mapsto 1/z$, we show
that these same zeros correspond to the scaled eigenvalues of a particular
multiplicative rank 1 perturbation of a random unitary matrix. Since the
correlation functions of the latter are known, by taking an appropriate limit
the correlation functions for the random Maclaurin series can be determined.
Only for $|\mu| \to \infty$ is a determinantal point process obtained. For the
one and two point correlations, by regarding the Maclaurin series as the limit
of a random polynomial, a direct calculation can also be given.
