| Summary: | This work continues the study of the properties of mutually homogeneous functions, which are a generalization of Euler homogeneous functions and can be used in the synthesis of electric and magnetic fields of electron and ion-optical systems with special properties. A chain of functions corresponding to multiple complex conjugate eigenvalues of the matrix of basic functional relations, is considered. Functional relations corresponding to such functions are derived. General formulas for the solutions of the obtained functional relations are derived. The properties of differentiability and integrability of the obtained functions are investigated. A generalization of the Euler theorem (Euler criterion) for the obtained class of differentiable functions is proved.
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