Numerically stable and computationally efficient expression for the magnetic field of a current loop

In this work it is demonstrated that a straightforward implementation of the commonly used textbook expressions of the magnetic field of a current loop is numerically unstable in a large region of interest. A novel expression based on a modification of Bulirsch’s algorithm for the complete elliptic integral is proposed. It is numerically stable and faster to compute than the classical one. New and old expressions are compared to series expansions and dipole approximation in terms of accuracy, numerical stability and computation performance.