We consider the problem of electrical conduction in the context of geophysical prospecting and assume that the conductivity of the Earth is constant in a direction perpendicular to the probing plane. The resulting boundary value problem is reduced to two dimensions via a Fourier transform with respect to this direction. To date, the typical method of solution involves solving several of these two-dimensional problems and computing the approximate inverse Fourier transform numerically. We propose a more efficient approach in which the inverse Fourier integral is taken analytically. This method involves the computation of an analytic function of the matrix approximation to a differential operator using its Lanczos decomposition. After deriving the method we present numerical verification of its validity and a discussion of its computational cost, which approaches that of two-dimensional problems.