Mutually homogeneous functions I. Matrices of the finite size

Abstract

This publication continues the study of the properties of Euler homogeneous functions that can be used in the synthesis of electric and magnetic fields of electron and ion-optical systems to carry out spectrographic mode record. A generalization of a functional equation of a general form for homogeneous functions is considered, which corresponds to linear functional relations with a matrix of minimal size. Under the assumption of differentiability of the functions under consideration, a general solution of the obtained functional equation is found. The resulting systems of functions are called mutually homogeneous functions by analogy with homogeneous Euler' functions and associated homogeneous Gel’fand’ functions. Further publications suggest the study of infinite matrices of functional relations corresponding to the fundamental chains of mutually homogeneous functions with real eigenvalues and complex conjugate eigenvalues.

Keywords

• functional equation
• associated homogeneous function
• mutually homogeneous functions
• spectrograph