Expansion of vacuum magnetic fields in toroidal harmonics

The vacuum magnetic field in stellarators and tokamaks is expanded in toroidal harmo ics (half-integer Legendre functions). In addition to the commonly used external harmonics (irregular at infinity), internal harmonics (irregular at the coordinate pole) are included in the expansion. This allows representation of the field from the central conductor in the TJ-II Heliac. The expansion is shown to provide a very accurate representation of the vacuum field in both cases. Algorithms for the accurate and rapid evaluation of the half-integer Legendre functions are provided. 1