The generalized ray expansion for the case of a structure containing any number of layers N welded on top of a half-space is presented, along with a rigorous mathematical proof based on the corresponding initial-boundary value problem. The main effort in obtaining the expansion consists in finding the series for the inverse of the secular function and bringing it to a convenient form. As a preliminary stage, the integrand of the potential in the transform domain is written explicitly in terms of the elements of the matrix product leading to the secular function. The expression for the potential is general in the sense that it includes all the possible source-receiver locations.
The expansion in generalized rays obtained here is expressed by an N-tuple sum depending on parameters representing the number of complete sections the corresponding ray travels within the various layers.