Finally, note that the winding number is closely related with the (2 + 1)-dimensional continuous Heisenberg ferromagnet equations and its integrable extensions: the Ishimori equation etc.
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Given that the function is analytic within each of these quarters, a nonzero winding number N (always an integer) identifies N zeros of the function inside the quarter in question by Rouché's theorem, each zero counted as many times as its multiplicity.