On the pole and zero locations of rational Laplace transformations of non-negative functions. II
A. H. Zemanian · Proceedings of the American Mathematical Society · 1961
In (2), the i and bi represent the real and complex zeros, respectively, and the pi and ti represent the real and complex poles, respectively. The complex poles and zeros will occur in complex conjugate pairs and the poles will have non positive real parts. Various sets of sufficient conditions on the poles and zeros, which insure that a(t) is nondecreasing, have been published previously [1-5]. The purpose of this note is to add to this body of results an extension of the theorem given in [5]. It includes as special cases most of the results given in [1; 2; and 5]. It does not, however, include the results of [3; 4]. The following notation and numbering system will be used for the poles and zeros. The real poles are numbered according to their decreasing values; that is, Pl >P2>* * * >Pm. The real parts of the complex poles and of all the zeros are denoted by axi and numbered according to their decreasing values; that is, a,i _ is2 > . . . >a,, where n=h+g+q. When several zeros and complex poles have the same real part, they are numbered in any order. If the multiplicity of any pole or zero is r, it is counted r times. The following fact is an immediate consequence of the first lemma and the last four sentences of [5].