Next, we define where is the entire even function defined in Theorem 2.8 Then is an entire function in the complex plane. Then we prove that converges as, and if we denote this limit by, one hasīut now is well defined for all In fact it is analytic in Where and the paths of the integrals are the segments joining the various points. Since is regular in the semi-strip, then for all so that, we define (to prove that, can be obtained similarly.)Īll three steps make use of the convergence of The rest of the proof is broken into three steps. This implies the existence of some such that $ f(z) $ is analytic for,. Then with no loss of generality we can also assume that this interval is where. Suppose now that there exists an interval of length greater than on the line on which has two singularity.
Abscissa of convergence series#
Thus, all three series converge absolutely and uniformly in any half-plane One also notes that from (12) we have With no loss of generality we assume that the abscissa of convergence (ordinary and absolute) is the line In other words the relation Since the are real positive numbers, we consider the non-trivial case, that is when the three series converge in identical half-planes of the form įrom Lemma 2.3, the regions of convergence of the three series are the same. We follow on the lines of the proof of Theorem XXIX in. Has at most one singularity in every interval of length exceeding on the abscissa of convergence. Then any Taylor-Dirichlet series as in (3), satisfying Let be a real positive sequence for Let so that is real positive too and let be its reordering. For any point inside the open convex region, the three series converge absolutely. The regions of convergence of the Taylor-Dirichlet series 29) proved that if is a multiplicity-sequence that satisfies the relations Let be a complex sequence satisfying and as Let and let be its reordering. (Phragmen-Lindelof.) Let be analytic in the region between two straight lines making an angle at the origin and on the lines themselves. Let be anyone of the following sequences:, , or where so that. There exist positive constants and so that for any n one has Similarly, if instead of a real sequence we have a complex sequence. Then the regions of convergence of the three series as defined inĪre the same. Let be a real positive sequence and let so that is real positive too, with its reordering. We state now lemmas that were proved in regarding multiplicity sequences. Observe that the disks in is not necessarily disjoint, since for fixed we might have for. We shall call this form of the reordering (see also ). We may now rewrite in the form of a multiplicity sequence, by grouping together all those terms that have the same modulus, and ordering them so that. Also note that allows for non-coinciding terms to come very close to each other. One observes that allows for the sequence to have coinciding terms. We say that a sequence with real positive terms, not necessarily in an increasing order, belongs to the class if for all we have Let the sequence and real positive numbers so that We denote by the class of all sequences with distinct complex terms diverging to infinity, satisfying the following conditions: (see also )ĭefinition 2.2. In this section, we describe the definitions and also to express and prove the lemma, we need to prove the theorem.ĭefinition 2.1. We note that other results concerning the location of singularities of Taylor–Dirichlet series have been derived by Blambert, Parvatham, and Berland (see ). We assume that the reader is familiar with the theory of Entire Functions and the theory of Dirichlet series, as used in the books. The Fabry Gap theorem () states that if, is a real positive sequence such that for and as, then the Dirichlet series has at least one singularity in every interval of length exceeding on the abscissa of convergence.