  A solution is proposed for finding the vector potential and magnetic field of
any distribution of currents with axial symmetry. In this approach, the
magnetic field and the vector potential are looked for not by solving a
differential equation but rather through straightforward calculation of
integrals of one scalar function. The solution is expressed in terms of the
associated Legendre polynomials P_{lm} with the index m of the Legendre
polynomials assuming one value only, m = 1. The solution has the form of a
series, with the coefficients of the polynomials being combinations of
multipole moments.
  Key words: electrodynamics, vector potential, spherical harmonics, Legendre
polynomials, magnetic field.
