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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Some interdependencies of sum and product integrals


Author: Jon C. Helton
Journal: Proc. Amer. Math. Soc. 37 (1973), 201-206
MSC: Primary 26A39
MathSciNet review: 0308340
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Abstract: In a recent paper Davis and Chatfield show if $ \smallint _a^b{G^2} = 0$, then $ \smallint _a^bG$ exists if and only if $ \prod\nolimits_a^b {(1 + G)} $ exists and is not zero. In this paper we extend that result and prove if $ \beta > 0,\vert G\vert < 1 - \beta $ on $ [a,b]$ and $ \smallint _a^b{G^2}$ exists, then $ \smallint _a^bG$ exists if and only if $ \prod\nolimits_a^b {(1 + G)} $ exists and is not zero. Furthermore, if $ \prod\nolimits_x^y {(1 + G)} = \exp \smallint _x^yG$ for $ a \leqq x < y \leqq b$, then $ \smallint _a^b{G^2} = 0$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1973-0308340-5
PII: S 0002-9939(1973)0308340-5
Keywords: Sum integral, product integral, interdependency, exponential, bound, interval function
Article copyright: © Copyright 1973 American Mathematical Society