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$J$-expanding mtrix functions and their role in the analytical theory of electrical circuits
A. V. Efimov,
V. P. Potapov
Abstract:
Chapter I establishes the essential properties of the
$\mathscr A$-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:
$J$-expanding matrix-functions (class
$\mathfrak M$), and also positive matrix functions (class
$\mathfrak B$ ), is introduced.
Chapter II gives an account of a theory of matrix functions of class
$\mathfrak M$. 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
$\mathfrak M$. In particular, the factorizability of a rational reactive matrix of class
$\mathfrak M$ is established.
Chapters III–IV set forth a theory of various subclasses of matrix functions of class
$\mathfrak M$:
$\mathfrak M_{sl}$,
$\mathfrak M_{cgl}$,
$\mathfrak M_{lr}$. The realizability of the matrix functions of each of these subclasses as
$\mathscr A$-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
$\mathfrak M_{clr}$ and
$\mathfrak M_{cglr}$ as
$\mathscr A$-matrices or
$z$-matrices of dissipative multipoles.
UDC:
519.53+512.83
MSC: 15A48,
15A15,
15A23