Direct algorithm for the rapid calculation of magnetic field spherical harmonics

Expressions for the direct calculation of the spherical harmonics of a magnetic field, generated by a circular current-carrying arc of rectangular cross section and arbitrary azimuthal length, are described. The final form contains a double integral with fixed range and no special functions. Solutions can be obtained rapidly and accurately by using Gauss-Legendre quadrature for numerical integration, and three-term recurrence relations for the calculation of associated Legendre polynomials. A comparison shows the distinct speed advantage this algorithm holds over an alternative, currently used technique, making the expressions ideal for inclusion in optimization problems, particularly with the explicit calculation of the magnetic field not required.