On the Restricted Validity of Point-Matching Techniques

The usefulness of point matching when the expansion representing the field diverges in the matching region is discussed in relation to previous work. It is shown that the presence of a metal boundary does not necessarily imply divergence of series representations at the boundary position. The Rayleigh hypothesis, which bears on this, and the extended-boundary-condition method are equivalent only in a restricted sense; the latter is hypersensitive to minute deviations of the field on the reduced boundary and is therefore unsuited to point matching for some shapes. A numerical example is examined in which the use of a divergent series gives little evidence of error from use in the divergent region. Attention is drawn to recent work in which the use of a sutliciently smooth spectral expansion is shown to be able to convert a series of divergent terms into a usable convergent series, permitting valid numerical computations with otherwise divergent representations.