×
Dawson, C. Bryan; Dawson, Matthew

Direct proofs of the fundamental theorem of calculus for the omega integral. (English) Zbl 1452.26023

Missouri J. Math. Sci. 31, No. 1, 46-55 (2019).
Summary: When introduced in a 2018 article in the American Mathematical Monthly, the omega integral was shown to be an extension of the Riemann integral. Although results for continuous functions such as the Fundamental Theorem of Calculus follow immediately, a much more satisfying approach would be to provide direct proofs not relying on the Riemann integral. This note provides those proofs.

MSC:

26E35 Nonstandard analysis

Keywords:

omega integral; fundamental theorem of calculus
PDFBibTeX XMLCite
\textit{C. B. Dawson} and \textit{M. Dawson}, Missouri J. Math. Sci. 31, No. 1, 46--55 (2019; Zbl 1452.26023)
Full Text: DOI arXiv Euclid

References:

[1] C. B. Dawson, A new extension of the Riemann integral, The American Mathematical Monthly, 125 (2018), 130-140, \ttdoi.org/10.1080/00029890.2018.1401832. · Zbl 1392.26039 · doi:10.1080/00029890.2018.1401832
[2] R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Graduate Texts in Mathematics, 188, Springer-Verlag, New York, 1998. · Zbl 0911.03032
[3] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964. · Zbl 0171.38503
[4] A. K. Agarwal, On a new kind of numbers, The Fibonacci Quarterly, 28.3 (1990), 194-199. · Zbl 0713.11015
[5] R. André-Jeannin, Generalized complex Fibonacci and Lucas functions, The Fibonacci Quarterly, 29.1 (1991), 13-18. · Zbl 0726.11009
[6] W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935. · JFM 61.0406.01
[7] P. S. Bruckman, On generating functions with composite coefficients, The Fibonacci Quarterly, 15.3 (1977), 269-275. · Zbl 0404.05005
[8] L. Carlitz, Note on irreducibility of Bernoulli and Euler polynomials, Duke Math. J., 19 (1952), 475-481. · Zbl 0047.25401
[9] L. Carlitz, Some identities of Bruckman, The Fibonacci Quarterly, 13.2 (1975), 121-126. · Zbl 0303.05012
[10] A. Erdélyi et al., Higher Transcendental Functions, Vol. 1, McGraw-Hill Book Company, Inc., New York, 1953. · Zbl 0051.30303
[11] W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third Edition, Springer-Verlag, Berlin, 1966. · Zbl 0143.08502
[12] E. D. Rainville, Special Functions, Chelsea Publ. Co., Bronx, New York, 1971. · Zbl 0231.33001
[13] L. · Zbl 0135.28101
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.