On the arguments of Jacobians in local expressions of the Maslov canonical operator

Local expressions for the Maslov canonical operator contain square roots of some Jacobians and, to correctly define the canonical operator, one must choose a branch of the root, i.e., in fact, a branch of the Jacobian argument, according to a certain rule. In most existing expositions of the theory of the canonical operator, this rule is one of the most mysterious places, at least if one tries to apply it in practice. In this paper, we remove the veil of mystery and present computationally efficient formulas suitable for computer implementation.