Wiener-Hopf factorization of piecewise meromorphic matrix-valued functions

Let $\mathrm D^+$ be a multiply connected domain bounded by a contour $\Gamma$, let $\mathrm D^-$ be the complement of $\mathrm D^+\cup\Gamma$ in $\overline{\mathbb C}={\mathbb C}\cup\{\infty\}$, and $a(t)$ be a continuous invertible matrix-valued function on $\Gamma$ which can be meromorphically extended into the open disconnected set $\mathrm D^-$ (as a piecewise meromorphic matrix-valued function). An explicit solution of the Wiener-Hopf factorization problem for $a(t)$ is obtained and the partial factorization indices of $a(t)$ are calculated. Here an explicit solution of a factorization problem is meant in the sense of reducing it to the investigation of finitely many systems of linear algebraic equations with matrices expressed in closed form, that is, in quadratures.
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