The notion and instrument of dispersion curves (DCs) and generalized DCs are widely used [1] as a form to establish a relation between a quantity (e.g. eigen- or natural frequencies) that governs resonances or normal waves (often it is a spectral parameter of the corresponding problem) or cloaking (when the suppression of several scattered-field harmonics takes place) and a non-spectral (complex) variable, e.g. permittivity. Generally, reliance of the propagation, resonance and cloaking phenomena on various quantities characterizing the scatterers and the incident field, e.g. the frequency dependence of the propagation constants of normal waves that gives rise to DCs, is considered in terms of the parameter dependence of one or several quantities on one or several parameters. In this respect, such parameter dependences are naturally connected with certain mappings (establishing relations between domains or manifolds) rather than with individual (families of) functions of one variable, as in the case of DCs. The present work formulates the prerequisites of this approach generalizing the DC method, taking the case when such mappings are specified as complex functions of a complex variable, and considering a homogeneous dielectric cylinder of circular cross-section as a reasonable basic example.

The approaches and ideas exploring the application of conformal mappings to the analysis of cloaking and invisibility proposed in [2] and surveyed in [3] have been used within a different ‘coordinate’ framework. One can find in [3] and the works cited therein several examples of mappings that can be useful also when the present method of investigating scattered-field expansion coefficients is applied.