<i>J</i>-EXPANDING MATRIX FUNCTIONS AND THEIR ROLE IN THE ANALYTICAL THEORY OF ELECTRICAL CIRCUITS

Abstract

Chapter I establishes the essential properties of the -matrix of a passive multipole depending on the number of its branches. These properties are based on Langevin's theorem.A classification of the basic objects of investigation: -expanding matrix-functions (class ), and also positive matrix functions (class ), is introduced.Chapter II gives an account of a theory of matrix functions of class . It also investigates the simplest (elementary and primary) matrices of this class. The fact is established that elementary (and primary) factors can be split off from a given matrix of class . In particular, the factorizability of a rational reactive matrix of class is established.Chapters III-IV set forth a theory of various subclasses of matrix functions of class : , , . The realizability of the matrix functions of each of these subclasses as -matrices of passive multipoles with the corresponding provision for branches is established.The fact that they are realizable is proved by the construction of a corresponding multipole.The last chapter is concerned with a generalization of Darlington's theorem, which leads to a realization of functions of the subclasses and as -matrices or -matrices of dissipative multipoles.