Missouri Journal of Mathematical Sciences

Direct Proofs of the Fundamental Theorem of Calculus for the Omega Integral

C. Bryan Dawson and Matthew Dawson

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Abstract

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.

Article information

Source
Missouri J. Math. Sci., Volume 31, Issue 1 (2019), 46-55.

Dates
First available in Project Euclid: 30 May 2019

Permanent link to this document
https://projecteuclid.org/euclid.mjms/1559181625

Digital Object Identifier
doi:10.35834/mjms/1559181625

Mathematical Reviews number (MathSciNet)
MR3960286

Subjects
Primary: 26E35: Nonstandard analysis [See also 03H05, 28E05, 54J05]

Keywords
omega integral fundamental theorem of calculus

Citation

Dawson, C. Bryan; Dawson, Matthew. Direct Proofs of the Fundamental Theorem of Calculus for the Omega Integral. Missouri J. Math. Sci. 31 (2019), no. 1, 46--55. doi:10.35834/mjms/1559181625. https://projecteuclid.org/euclid.mjms/1559181625


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References

  • 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.
  • R. Goldblatt, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, Graduate Texts in Mathematics, 188, Springer-Verlag, New York, 1998.
  • M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.
  • A. K. Agarwal, On a new kind of numbers, The Fibonacci Quarterly, 28.3 (1990), 194–199.
  • R. André-Jeannin, Generalized complex Fibonacci and Lucas functions, The Fibonacci Quarterly, 29.1 (1991), 13–18.
  • W. N. Bailey, Generalized Hypergeometric Series, Cambridge University Press, Cambridge, 1935.
  • P. S. Bruckman, On generating functions with composite coefficients, The Fibonacci Quarterly, 15.3 (1977), 269–275.
  • L. Carlitz, Note on irreducibility of Bernoulli and Euler polynomials, Duke Math. J., 19 (1952), 475–481.
  • L. Carlitz, Some identities of Bruckman, The Fibonacci Quarterly, 13.2 (1975), 121–126.
  • A. Erdélyi et al., Higher Transcendental Functions, Vol. 1, McGraw-Hill Book Company, Inc., New York, 1953.
  • W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, Third Edition, Springer-Verlag, Berlin, 1966.
  • E. D. Rainville, Special Functions, Chelsea Publ. Co., Bronx, New York, 1971.
  • L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.