We derive a set of functions which satisfies Laplace's equation in two dimensions x and y, when the longitudinal or z axis has a constant but nonzero curvature. The functions generalize the Fourier harmonics of the standard two-dimensional multipole expansion. Both the electrostatic scalar potential and the magnetic vector potential, and the electric and magnetic fields, are treated. Recursion relations are also supplied, in a form suitable for use in a computer program, to evaluate the multipoles to arbitrary order. A comparison with other work in the field is also presented.