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We derive a determinant formula for the WKB exponential of singularlyperturbed Zakharov-Shabat system that corresponds to the semiclassical (zerodispersion) limit of the focusing Nonlinear Schr\" odinger equation. Thederivation is based on the Riemann-Hilbert Problem (RHP) representation of theWKB exponential. We also prove its independence of the branchpoints of thecorresponding hyperelliptic surface assuming that the modulation equations aresatisfied.

Determinant form of modulation equations for the semiclassical focusing Nonlinear Schr\" odinger equation