This paper is concerned with the two-dimensional fields generated by time harmonic compression and shear line sources in a semi-infinite homogeneous isotropic elastic solid. It is well known that these elastodynamic states may easily be expressed as Fourier integral superpositions of plane waves in cartesian coordinates over a continuous spectrum of wavenumbers. The problem addressed here is that of determining the analytic form of these expressions in the neighbourhood of their singular points. The natural coordinate system for the problem is a cylindrical polar system centred at those singular points and thus the Fourier integrals are expanded in terms of cylindrical wave functions. The expansion method presented here is not special to the elasticity problem and has numerous important analytical applications in wave scattering problems in homogeneous and composite bodies.