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This paper is concerned with the Laplace boundary-value problem with thedirectional derivative, corresponding to the specific nature of measurements ofthe longitudinal component of the photospheric magnetic field. Boundaryconditions are specified by a distribution on the sphere of projection of themagnetic field vector unto a given direction. It is shown that the solution ofthis problem exists in the form of a spherical harmonic expansion, anduniqueness of this solution is proved. A conceptual sketch of numericaldetermination of the harmonic series coefficients is given. The field ofapplication of the method is analyzed having regard to the peculiarities ofactual data. Finally, we present differences in results derived fromextrapolating the magnetic field from a synoptic map and a full-diskmagnetogram.Comment: 18 pages, 2 Gif figures, uses kluwer.sty, submitted to Solar Phys, I  have also a Russian version in the form Preprint: ISTP SD RAS, 1999, No.  2-99, 14 p. (rud@iszf.irk.ru

Extrapolation of the solar magnetic field within the potential-field approximation from full-disk magnetograms