  Recently, D. Wang has devised a new contour integral based method to simplify
certain matrix integrals. Capitalizing on that approach, we derive a new
expression for the probability density function (p.d.f.) of the joint
eigenvalues of a complex non-central Wishart matrix with a rank-1 mean. The
resulting functional form in turn enables us to use powerful classical
orthogonal polynomial techniques in solving three problems related to the
non-central Wishart matrix. To be specific, for an $n\times n$ complex
non-central Wishart matrix $\mathbf{W}$ with $m$ degrees of freedom ($m\geq n$)
and a rank-1 mean, we derive a new expression for the cumulative distribution
function (c.d.f.) of the minimum eigenvalue ($\lambda_{\min}$). The c.d.f. is
expressed as the determinant of a square matrix, the size of which depends only
on the difference $m-n$. This further facilitates the analysis of the
microscopic limit for the minimum eigenvalue which takes the form of the
determinant of a square matrix of size $m-n$ with the Bessel kernel. We also
develop a moment generating function based approach to derive the p.d.f. of the
random variable $\frac{\text{tr}(\mathbf{W})}{\lambda_{\min}}$, where
$\text{tr}(\cdot)$ denotes the trace of a square matrix. This random quantity
is of great importance in the so-called smoothed analysis of Demmel condition
number. Finally, we find the average of the reciprocal of the characteristic
polynomial $\det[z\mathbf{I}_n+\mathbf{W}],\; |\arg z|<\pi$, where
$\mathbf{I}_n$ and $\det[\cdot]$ denote the identity matrix of size $n$ and the
determinant, respectively.
