The electric and magnetic fields that an overhead infinitely long line current produces at the surface of the Earth can be expressed as inverse Fourier integrals over a horizontal wave number. In the quasi-static approximation acceptable in most applications and assuming a uniform Earth, expressions in terms of the Neumann and Struve functions are also possible. These are further developed in this paper, resulting in new infinite series expansions, not presented before, for the electric and magnetic fields. An application of the series expansions will remove the need to calculate the standard integral expressions, and performing exact numerical computations will be straightforward. ¿ 1998 American Geophysical Union