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Explicit soft fringe maps of a quadrupole

The analytic linear and nonlinear zero-length fringe maps are calculated explicitly for a normal quadrupole up to 6-th order generators. A very simple leading term is found for the linear fringe map. The 4-th order leading term is the L. Whiting's hard edge result (1970) as expected. No significant leading term at the 6-th order are found. Our results should be useful to estimate the significancy of the soft fringe of a quadrupole to beam dynamics. The method used in this calculation can be used to compute the soft fringe maps of various magnets.