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In recent years a vast amount of literature has been written on singularity and eigenmode expansion methods (SEM and EEM). The main practical purpose is to produce a theory which allows us to identify a flying (or stationary) target from the observed transient field scattered by the target. (The other purposes include representation and computation of the transient field and construction of equivalent circuits according to the object's electromagnetic response.) This problem is discussed. Described are 1) the basic starting points of the engineering approach to the problem and the extent to which they are consistent from the mathematical point of view, 2) the material which has been rigorously established in the field, 3) the important points in practice, and 4) the unsolved mathematical problems in the field. This work may save efforts by other scientists and engineers, since the results demonstrate the kind of research that can be used in practice and the kind which is of mostly theoretical interest.