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An exact solution is given for the fields of an infinitely permeable asymmetrical finite-pole-tip ring head. The solution is obtained by conformal mapping and is expressed in terms of elliptic integrals and elementary functions. Using the method of undetermined coefficients, it is shown how to decompose the Schwarz-Christoffel integral into elementary functions and elliptic integrals. For each given pole geometry, the constants appearing in the Schwarz-Christoffel formula are calculated once. Then the fields at various points in the poles domain are calculated using Newton's method and the formulas for the decomposition of the Schwarz-Christoffel integral.