Singularities of the continuation of fields and validity of Rayleigh’s hypothesis

To formulate general results concerning the validity of the Rayleigh hypothesis, we first introduce a definition of the foci and antifoci of an analytic curve. Then, we state two lemmas on the properties of an analytic or harmonic function satisfying given conditions on an analytic curve. This allows us to predict the behavior of the analytic continuation of the field in electrostatics. The use of a conformal mapping permits the generalization of this method in electromagnetics and acoustics. As a consequence, we are able to predict the limit of validity of the Rayleigh hypothesis.