Authors:
Ferenc Móricz University of Szeged Bolyai Institute Aradi Vértanúk Tere 1 6720 Szeged Hungary

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Ф. Мориц University of Szeged Bolyai Institute Aradi Vértanúk Tere 1 6720 Szeged Hungary

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Abstract  

First, we consider integrals of the form

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\int {a(x)m^{ - x} dx for} m = 2,3,...$$ \end{document}
over the unit interval (0, 1) or the interval (1, ∞) or the half-line (0, ∞), wherea(x)≥0 and is integrable on the interval in question. These integrals are related to the Dirichlet series
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\sum\limits_{m = 2}^\infty {a_m m^{ - x} for x > 1} ,$$ \end{document}
, where the numbersam≥0. We survey certain known results in a new formulation in order to reveal the difference in behavior between the functions which are integrable on either (0, 1) or (1, ∞). Their proofs can be read out from the existing literature. Second, we extend these results from single to double related integrals, while making distinction among the functionsa(x, y) which are integrable on either (0, 1)2 or (0, 1)×(1, ∞) or (1, ∞)×(0, 1) or (1, ∞)2. The case wherea(x, y) is integrable on (0, ∞)2 is also included.

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Analysis Mathematica
Language English
Size B5
Year of
Foundation
1975
Volumes
per Year
1
Issues
per Year
4
Founder Akadémiai Kiadó
Founder's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245
Publisher Akadémiai Kiadó
Springer Nature Switzerland AG
Publisher's
Address
H-1117 Budapest, Hungary 1516 Budapest, PO Box 245.
CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible
Publisher
Chief Executive Officer, Akadémiai Kiadó
ISSN 0133-3852 (Print)
ISSN 1588-273X (Online)