The solution of Laplace's equation, in spherical coordinates, is developd for the boundary value problem appropriate to fitting the geomagnetic field over a spherical cap. The solution involves associated Legendre functions of integral order but nonintegral degree. The basis functions comprise two infinite sets, within each of which the functions are mutually orthogonal. The series for the expansion of the potential can be design be differentiated term by term to yield uniformly convergent series for the field components. The method is demonstrated by modeling the International Geomagnetic Reference Field 1980 at the earth's surface and upward continuing it to 300 and 600 km. The rate of convergence of the series is rapid, and standard errors of fit as low as the order of a nanotesla can be obtained with a reasonable number of coefficients. Upward continuation suffers from not considering data outisde the cap, the deterioration being confined to the boundary at low continuation altitudes but spreading inward over the cap with increasing altitudes. At 600 km the standard error of upward continuation is about 5 times the standard error of fit. Both the fit and the upward continuation can be greatly improved at a given truncation level by subtraction of a known spherical harmonic potential determined from data from the whole earth.