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How to Compute Antiderivatives

Published online by Cambridge University Press:  15 January 2014

Chris Freiling
Chris Freiling*
Affiliation:
Mathematics Department, California State University, San Bernardino, California 92407. E-mail: [email protected]
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Extract

This is not about the symbolic manipulation of functions so popular these days. Rather it is about the more abstract, but infinitely less practical, problem of the primitive. Simply stated:

Given a derivative f: ℝ → ℝ, how can we recover its primitive?

The roots of this problem go back to the beginnings of calculus and it is even sometimes called “Newton's problem”. Historically, it has played a major role in the development of the theory of the integral. For example, it was Lebesgue's primary motivation behind his theory of measure and integration. Indeed, the Lebesgue integral solves the primitive problem for the important special case when f(x) is bounded. Yet, as Lebesgue noted with apparent regret, there are very simple derivatives (e.g., the derivative of F(0) = 0, F(x)= x2 sin(1/x2)for x ≠ 0) which cannot be inverted using his integral.

The general problem of the primitive was finally solved in 1912 by A. Denjoy. But his integration process was more complicated than that of Lebesgue. Denjoy's basic idea was to first calculate the definite integral f(x) dx over as many intervals (a,b) as possible, using Lebesgue integration. Then, he showed that by using these results, the definite integral could be found over even more intervals, either by using the standard improper integral technique of Cauchy, or an extension technique developed by Lebesgue (see appendix for details).

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 1 , Issue 3 , June 1995 , pp. 279 - 316
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

[1] Blass, A. and Cenzer, D., Cores of sets of reals, Journal of Symbolic Logic, vol. 39 (1974), pp. 649–654.CrossRefGoogle Scholar
[2] Bruckner, A., Differentiation of real functions, CRM Monograph Series, no. 5, American Mathematical Society, Providence, 1994.CrossRefGoogle Scholar
[3] Bullen, P. S., Non-absolute integrals: a survey, Real Analysis Exchange, vol. 5 (197980), pp. 195–259.CrossRefGoogle Scholar
[4] Cenzer, D. and Mauldin, R. D., Inductive definability: measure and category, Advances in Mathematics, vol. 38 (1980), no. 1, pp. 55–90.CrossRefGoogle Scholar
[5] Darji, U., Evans, M., and O'Malley, R., First return path systems: differentiability, continuity and orderings, to appear.Google Scholar
[6] Denjoy, A., Calcul de la primitive de la fonction derivée la plus genérale, Comptes rendus hebdomadaires des séanses de l' Académie des sciences Paris, Series I, Mathematiques, vol. 154 (1912), pp. 1075–1078.Google Scholar
[7] Denjoy, A., Leçons sur le calcul des coefficients d'une série trigonométrique, i-iv, Paris, 1941–1949.Google Scholar
[8] Dougherty, R. and Kechris, A.S., The complexity of antidifferentiation, Advances in Mathematics, vol. 88 (1991), pp. 145–169.CrossRefGoogle Scholar
[9] Harel, D. and Kozen, D., A programming language for the inductive sets, and applications, Information and Control, vol. 63 (1984), pp. 118–139.CrossRefGoogle Scholar
[10] Kleene, S., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 405–428.CrossRefGoogle Scholar
[11] Looman, H., Über die Perronsche Integraldefinition, Mathematische Annalen, vol. 93 (1925), pp. 153–156.CrossRefGoogle Scholar
[12] Matiyasevich, Y., A new proof of the theorem on exponential diophantine representation of enumerable sets, Journal of Soviet Mathematics, vol. 14 (1980), pp. 1475–1486.CrossRefGoogle Scholar
[13] Moschovakis, Y. N., Elementary induction on abstract structures, North-Holland, Amsterdam, 1975.Google Scholar
[14] Moschovakis, Y. N., Descriptive set theory, North-Holland, Amsterdam, 1980.Google Scholar
[15] Natanson, I. P., Theory of functions of a real variable, Ungar, New York, 1955 and 1959.Google Scholar
[16] Pesin, I., Classical and modern integration theories, Academic Press, New York and London, 1970.Google Scholar
[17] Rogers, H., Theory of recursive functions and effective computability, McGraw-Hill Series in Higher Mathematics, McGraw-Hill, New York, 1967.Google Scholar
[18] Saks, S., Theory of the integral, Hafner Publishing Company, Warsaw, 1937.Google Scholar
[19] Zahorski, Z., Sur la primière dérivée, Transactions of the American Mathematical Society, vol. 69 (1950), pp. 1–54.Google Scholar