| 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 real eigenvalues of the matrix of basic functional relations for mutually homogeneous functions, is considered. Functional relations corresponding to such functions are derived. General formulas for the solutions of the obtained functional relations are derived. It is shown that the obtained functions are a refinement of the associated Gel’fand functions. 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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