Expressions for the electric and magnetic vector potentials for z-directed magnetic and electric current point sources are developed as spectral integrals over longitudinal and azimuthal wave numbers. The representations are valid outside the source region. The construction is systemized, with the aid of Mason's signal-flow graphs, in order to facilitate numerical implementation. The azimuthal spectral integral form is chosen rather than the azimuthally periodic form so that computations may be carried out in the complex plane associated with angular wave number without recourse to the Watson transform. Some example calculations are provided to demonstrate the practicability of the method.