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The limiting behavior of the eigenvalues of the Toeplitz matrices$T_{n}[\sigma]=(\hat{\sigma}(i-j))$, where $0\leq i,j \leq n$, as $n \to\infty$, is investigated in the case of complex valued functions $\sigma$defined on the unit circle $\mathbb{T}$ and having exactly one point ofdiscontinuity. It is found that if $\sigma(z)=(-z)^{\beta}\tau(z)$, $\beta$ notan integer and $\tau$ satisfying certain smoothness conditions, then $\detT_{n}[\sigma]=\mathbf{G}[\tau]^{n+1}n^{-\beta^{2}}E[\tau,\beta](1+o(1))$ as $n\to \infty$, where $\mathbf{G}[\tau]$ denotes the geometric mean of $\tau$ and$E$ is a constant independent of $n$. A value for $E$ is found in terms of theFourier coefficients of $\tau$ and an analytic function of $\beta$. Theseresults were known previously in the case that $\Re \beta$, the real part of$\beta$, was sufficiently small. A corollary of this result is a determinationof the limiting set and limiting distributions for the eigenvalues of$T_{n}[\sigma]$.

Spectral Asymptotics for Toeplitz Matrices Having Certain Piecewise Continuous Symbols