Abstract
From after what has been said in the previous lecture, the indefinite integral
is nothing other than the general value of the unknown y, subject to satisfy the differential equation
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Notes
- 1.
Cauchy continues to stress the fact his arbitrary constant is a piecewise constant function of x, along with its inherent points of discontinuity.
- 2.
What must be a typographical error occurs in the text of the 1899 edition. It replaces the 1823 edition’s intégrales indéfinies with the phrase intégrales définies, which would change the translation from an indefinite to a definite integral. Cauchy’s originally intended, and appropriate, 1823 phrase is used here.
- 3.
Precisely in the form we commonly see integration by parts today.
- 4.
One might picture Cauchy proving \(0=1\) to his students with this idea, perhaps interjecting a bit of a riddle and some humor into his highly intellectual and academic class – Cauchy himself hoping for a rare chuckle from his students.
Throughout this text, Cauchy occasionally makes assumptions about characteristics of his functions which in general may not be true. During the latter part of the 19th century, several pathological functions were generated which tested the new rigorous calculus theory. Although these functions may have taken Cauchy by surprise in 1823, the purpose was not to discredit the emerging theory. Instead, their introduction was constructive and used to locate weaknesses or ambiguities in the foundation of the subject, so the theory could be strengthened. Among these are the Weierstrass Function, a real valued function designed by Karl Weierstrass that is continuous everywhere, but differentiable nowhere. This unusual function clearly demonstrates intuition cannot be trusted as this function shows continuity is not enough to guarantee differentiability, as Cauchy tends to do. This particular function would have been unimaginable to Cauchy in 1823. Riemann developed a similarly bizarre function which is continuous everywhere, yet not differentiable almost everywhere. Another classic pathological example is the Indicator Function, sometimes referred to as the Dirichlet Function, named after Lejeune Dirichlet (1805–1859). It is a strange function that is nowhere continuous. A final example is the Ruler Function, developed by Carl Johannes Thomae (1840–1921). It contains an infinite number of discontinuities yet remains integrable and is continuous almost everywhere. These functions, along with many others, helped uncover and then strengthen the foundation of modern calculus. Pretty cool stuff.
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Cates, D.M. (2019). VARIOUS PROPERTIES OF INDEFINITE INTEGRALS. METHODS TO DETERMINE THE VALUES OF THESE SAME INTEGRALS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_27
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