Search

In the paper, series of exponential monomials are considered. We study the problem of the distribution of singular points of the sum of a series on the boundary of its domain of convergence. We study the conditions under which, for any sequence of coefficients of the series with a chosen domain of convergence, the domain of existence of the sum of this series coincides with the given domain of convergence. We consider sequences of exponents having an angular density (measurable) and the zero condensation index. Various criteria related to the distribution of singular points of the sum of a series of exponential monomials on the boundary of its convergence domain are obtained. In particular, in the class of the indicated sequences, a criterion is obtained that all boundary points of a chosen convex domain are special for any sum of a series with a given domain of convergence. The criteria are formulated using simple geometric characteristics of the sequence of exponents and a convex domain (the angular density and the length of the boundary arc). It is also shown that the condition that the condensation index is equal to zero is essential. [ABSTRACT FROM AUTHOR]

Let ( λ n ) be a sequence of the pairwise distinct complex numbers. For a formal Dirichlet series F ( z ) = + ∞ ∑ n = 0 a n e z λ n, z ∈ C, we denote G μ ( F ), G c ( F ), G a ( F ) the domains of the existence, of the convergence and of the absolute convergence of maximal term μ ( z, F ) = max { | a n | e R ( z λ n ) : n ≥ 0 }, respectively. It is well known that G μ ( F ), G a ( F ) are convex domains. Let us denote N 1 ( z ) := { n : R ( z λ n ) > 0 }, N 2 ( z ) := { n : R ( z λ n ) < 0 } and α ( 1 ) ( θ ) := lim –––– n → + ∞ n ∈ N 1 ( e i θ ) − ln | a n | R ( e i θ λ n ), α ( 2 ) ( θ ) := ¯¯¯¯¯¯¯¯ lim n → + ∞ n ∈ N 2 ( e i θ ) − ln | a n | R ( e i θ λ n ) . Assume that a n → 0 as n → + ∞ . In the article, we prove the following statements. 1 ) If α ( 2 ) ( θ ) < α ( 1 ) ( θ ) for some θ ∈ [ 0, π ) then { t e i θ : t ∈ ( α ( 2 ) ( θ ), α ( 1 ) ( θ ) ) } ⊂ G μ ( F ) as well as { t e i θ : t ∈ ( − ∞, α ( 2 ) ( θ ) ) ∪ ( α ( 1 ) ( θ ), + ∞ ) } ∩ G μ ( F ) = ∅ . 2 ) G μ ( F ) = ⋃ θ ∈ [ 0, π ) { z = t e i θ : t ∈ ( α ( 2 ) ( θ ), α ( 1 ) ( θ ) ) } . 3 ) If h := lim –––– n → + ∞ − ln | a n | ln n ∈ ( 1, + ∞ ), then ( h h − 1 ⋅ G a ( F ) ) ⊃ G μ ( F ) ⊃ G c ( F ) . If h = + ∞ then G a ( F ) = G c ( F ) = G μ ( F ), therefore G c ( F ) is also a convex domain. [ABSTRACT FROM AUTHOR]

In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling’s formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica.

Euler's reputation is tarnished because of his views on divergent series. He believed that all series should have a value, not necessarily a limit as for convergent series, and that the value should remain invariant irrespective of the method of evaluation. Via the key concept of regularisation, which results in the removal of the infinity in the remainder of a divergent series, regularised values can be evaluated for elementary series outside their circles of absolute convergence such as the geometric series and for more complicated asymptotic series called terminants. Two different techniques for evaluating the regularised values are presented: the first being the standard technique of Borel summation and the second being the relatively novel, but more powerful, Mellin-Barnes regularisation. General forms for the regularised values of the two types of terminants, which vary as the truncation parameter is altered, are presented using both techniques over the entire complex plane. Then an extremely accurate and extensive numerical study is carried out for different values of the magnitude and argument of the main variable and the truncation parameter. In all cases it is found that the MB-regularised forms yield identical values to the Borel-summed forms, thereby vindicating Euler's views and restoring his status as perhaps the greatest of all mathematicians. [ABSTRACT FROM AUTHOR]

In this work the complete version of Stirling's formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma function over all branches of the complex plane. Exact values can only be obtained by regularization. Two methods are introduced: Borel summation and Mellin–Barnes (MB) regularization. The Borel-summed remainder is composed of an infinite convergent sum of exponential integrals and discontinuous logarithmic terms that emerge in specific sectors and on lines known as Stokes sectors and lines, while the MB-regularized remainders reduce to one complex MB integral with similar logarithmic terms. As a result that the domains of convergence overlap, two MB-regularized asymptotic forms can often be used to evaluate the logarithm of the gamma function. Though the Borel-summed remainder has to be truncated, it is found that both remainders when summed with (1) the truncated asymptotic series, (2) Stirling's formula and (3) the logarithmic terms arising from the higher branches of the complex plane yield identical values for the logarithm of the gamma function. Where possible, they also agree with results from Mathematica. [ABSTRACT FROM AUTHOR]