  We explore the analytic structure of the three-channel $S$ matrix by
generalizing uniformization and making a single-valued map for the
three-channel $S$ matrix. First, by means of the inverse Jacobi's elliptic
function we construct a transformation from eight Riemann sheets of the
center-of-mass energy squared complex plane onto a torus, on which the
three-channel $S$ matrix is represented single-valued. Secondly, we show that
the Mittag-Leffler expansion, a pole expansion, of the three-channel scattering
amplitude includes not only topologically trivial but also nontrivial
contributions and is given by the Weierstrass zeta function. Finally, we
examine the obtained formula in the context of a simple three-channel model.
Taking a simple non-relativistic effective field theory with contact
interaction for the $S=-2$, $I=0$, $J^P = 0^+$,
$\Lambda\Lambda-N\Xi-\Sigma\Sigma$ coupled-channel scattering, we demonstrate
that the scattering amplitude as a function of the uniformization variable is,
in fact, given by the Mittag-Leffler expansion with the Weierstrass zeta
function and that it is dominated by contributions from neighboring poles.
