Augustin-Louis Cauchy and Cours d'Analyse Darius Ansari 10/06/2014 Augustin-Louis Cauchy was a mathematician born in Paris, France in the year 1789, just one month after the beginning of the French Revolution. After his family survived the revolution as well as the following Reign of Terror, Cauchy's father worked directly underneath great mathematician Laplace, and was a friend to Lagrange as well. Augustin-Louis Cauchy was at first a civil engineer, but he would soon be best known for his numerous contributions to complex functions and calculus, the latter of which will be explored in futher detail. One of Cauchy's works includes Cours d'Analyse de l'Ecole Royale Poly- technique or, more simply, Cours d'Analyse. Cours d'Analyse is a textbook in infinitesimal calculus published in 1821. Interestingly enough, Cauchy states in his textbook that (translated to english): As for the methods, I have sought to give them all the rigor which one demands from geometry, so that one need never rely on arguments drawn from the generality of algebra. It is for this reason{and his subsequent work, of course{that Cauchy is well- recognized for his rigorous analysis of calculus, including some of his own definitions, proofs, and methods. In Cauchy's "Preliminaries" to his Cours d'Analyse, he begins by provid- ing verbal definitions for numbers, quantities, sums, differences, and other operations. Of our particular interest, however, he defines the variable: We call a quantity variable if it can be considered as able to take on successively many different values. We normally denote such a quantity by a letter taken from the end of the alphabet. He almost immediately continues with the definition: When the values successively attributed to a particular variable indefinitely approach a fixed value in such a way as to end up 1 differing from it by as little as we wish, this fixed value is called the limit of all the other values. Thus, for example, an irrational number is the limit of the various fractions that give better and better approximations to it. And finally, When the successive numerical values of a such a variable de- crease indefinitely, in such a way as to fall below any given number, this variable becomes what we call infinitesimal, or an infinitesimally small quantity. A variable of this kind has zero as its limit. One can easily infer from Cauchy's explicit definitions that he was precise in his calculus-related work, and with these definitions out of the way, he begins to work with functions, their continuity, and other aspects of calculus as we know today. For example, section "2.2 On the continuity of functions." states: The function f(x) is continuous with respect to x between the given limits if, between these limits, an infinitely small increment in the variable always produces an infinitely small increment in the function itself. Or in other words, if f(x) is a function of the variable of x, and the function takes a unique finite value for each value of x between two given limits, then an infinitely small increment α added to x increments the function by the difference f(x + α) − f(x). f(x) is then continuous if the numerical value of the difference f(x−α)−f(x) decreases indefinitely with the numerical value of α. Following from his provided definitions of continuity, Cauchy states the Intermediate Value Theorem. Cauchy has also stated an alternate version of the Mean Value Theorem, known as the Extended Mean Value Theorem, as it is a generalization of the former. Augustin-Louis Cauchy used ,δ arguments in his proofs, notably in his definition of the derivative. On the other hand, Cauchy's name is associated with a term in sequences and series; the Cauchy sequence. The definition: 2 A sequence is called a Cauchy sequence if the terms of the sequence eventually all become arbitrarily close to one another. That is, given > 0 there is a positive integer N such that if m, n > N then jam − anj < . In fact, some of Cauchy's greatest work in calculus concerns sequences and series. He was the first to prove Taylor's theorem in a rigorous manner, and his textbook on the subject also includes his tests for absolute convergence that today bear his name. For the Taylor polynomial 00 k 0 f (a) f (a) f(x) = f(a) + f (a)(x − a) + (x − a)2 + ::: + (x − a)k 2! k! (where f has k continuous derivatives at x = a) the Cauchy form of the remainder is: (f k+1)(ξ ) R (x) = C (x − ξ )k(x − a) k (k!) C for some real number ξC between a and x. The Taylor polynomial approxi- mates f(x) near a, and Taylor's Theorem provides the bounds for the error. All in all, there have been numerous concepts throughout mathematics named for Augustin-Louis Cauchy. This is well-justified by Cauchy's great life and contributions, including hundreds of articles and a few complete textbooks such as Cours d'Analyse. Cauchy held many university positions and spent much of his life writing and publishing. He has made signifi- cant contributions to areas of mathematics and science such as calculus, complex functions, elasticity, group theory, hydrodynamics, and differential equations. However, Cauchy will forever be praised for providing one of the first reasonably successful rigorous foundations for calculus, and such appreciation is reflected in today's understanding of limits, continuity, and Taylor polynomials. 3.