Abstract
If, in equation (23) of the preceding lecture, we replace f(z) by the value \( F^{(n)}(z) \) derived from formula (21), we will find, under the same conditions.
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- 1.
Here Cauchy has expressed Taylor’s Theorem with an integral remainder term. The basic idea of Taylor’s Theorem had been stated as early as 1671 by James Gregory (1638–1675), but it is named after Brook Taylor (1685–1731), an English mathematician who originally wrote something close in the early 18th century. However, the theorem did not become widely known until Leonhard Euler used the result in his work several decades later. The theorem was not more fully developed to include a remainder term until much later by Lagrange and Cauchy.
- 2.
The definite integral term in equation (3).
- 3.
Gaspard de Prony (1755–1839) and Cauchy shared a mutual distaste for one another, dating back to when Cauchy was a student in de Prony’s mechanics class at the École Polytechnique. Unfortunately for Cauchy, much of the period in which Cauchy himself was teaching at the École Polytechnique, de Prony was responsible for overseeing the quality of Cauchy’s teaching. During Cauchy’s tenure, it is clear de Prony was generally dissatisfied with what he observed, feeling much of Cauchy’s teachings were too abstract. Mr. de Prony would continue producing unfavor-able reports of Cauchy up until the time Cauchy leaves the school in 1830. To make matters even worse for Cauchy, de Prony was also a highly influential member of the Conseil d’Instruction (the committee responsible for overseeing the curriculum) while Cauchy was battling over the content of his analysis course at the École Polytechnique, which created additional friction. It must have been a nightmare for Cauchy.
- 4.
The Fundamental Theorem of Calculus.
- 5.
The general form of the Mean Value Theorem for Definite Integrals with \(\chi (z)=\frac{(x-z)^{n-1}}{1\cdot 2\cdot 3\cdots (n-1)}\) and \(\varphi (z)=F^{(n)}(z).\)
- 6.
Equation (12) is one of the more common forms to express Taylor’s Theorem with remainder. The remainder in this case is called the Lagrange form. Cauchy has shown as long as the first n derivatives of f(x) are continuous in the neighborhood of x, there exists a \(\theta \) between zero and one in which this expansion is valid—essentially one way of stating the Taylor’s Theorem of today. He has now completed an integral development of Taylor’s Theorem. Beautiful.
- 7.
Cauchy adds a final derivation of Taylor’s formula in his ADDITION at the end of his original text. In this later derivation, he develops the formula without the integration techniques employed in this and previous lectures, using only differential calculus methods and established theorems.
- 8.
Cauchy is referring to his first serious inquiry into how to resolve the true value of the indeter-minate fraction \(\frac{0}{0}.\) The solution offered in Lecture Six determines the ratio of the first derivatives. However, this Lecture Six solution fails if this new ratio is also found to be indeterminate. Here, Cauchy looks to the ratio of higher and higher order derivatives until a determinate ratio is located.
Although it would have certainly cleaned up many of the expressions in the latter lectures of Cauchy’s text, the factorial notation we use today, m!, to indicate the operation \(1 \cdot 2 \cdot 3 \cdots m\) was not in common use at the time Cauchy wrote Calcul infinitésimal. Christian Kramp (1760–1826), a French mathematician, is credited with the introduction of the modern notation as early as 1808; however, it was not widely adopted until later in the century.
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Cates, D.M. (2019). TRANSFORMATION OF ANY FUNCTIONS OF x OR OF \(x+h\) INTO ENTIRE FUNCTIONS OF x OR OF h TO WHICH WE APPEND DEFINITE INTEGRALS. EQUIVALENT EXPRESSIONS TO THESE SAME INTEGRALS.. In: Cauchy's Calcul Infinitésimal. Springer, Cham. https://doi.org/10.1007/978-3-030-11036-9_36
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DOI: https://doi.org/10.1007/978-3-030-11036-9_36
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