A new method of describing the magnetic fringing field is proposed. While the traditional method gives the two-dimensional magnetic field distribution by terminating the summation of the Maclaurin expansions at a finite derivative order, we summed the expansions up to infinity using the complex variable method. The exact summation shows an occurrence of singularities to which no attention has been paid. Because the traditional empirical function causes many unnecessary singularities where there should not be in the two-dimensional space, we derived a new empirical function whose singularities are located where they should be. The new function can reproduce the two-dimensional field distribution seamlessly and very precisely in the full region where we need the field information. © 2004 Elsevier B.V. All rights reserved.