Augustin Louis Cauchy ( ) Laplace and Lagrange were visitors at the Cauchy family In1805 he took the entrance examination for the École Polytechnique. He was examined by Biota and placed second. At the École Polytechnique his analysis tutor was Ampère. In 1807 he graduated from the École Polytechnique and entered the engineering school École des Ponts et Chaussées. In 1810 Cauchy took up his first job in Cherbourg to work on port facilities for Napoleon's English invasion fleet In 1816 he won the Grand Prix of the French Academy of Science for a work on waves. He achieved real fame however when he submitted a paper to the Institute solving one of Fermat's claims on polygonal numbers made to Marlene lectured on methods of integration at the Collège de France. His text Cours d'analyse in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. In 1831 Cauchy went to Turin and after some time there he accepted an offer from the King of Piedmont of a chair of theoretical physics. In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor his grandson. Met with Bolzano. Cauchy returned to Paris in 1838 and regained his position at the Academy Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences. He produced 789 mathematics papers,

Augustin Louis Cauchy ( ) Lectures on the Infinitesimal Calculus. First Lesson: Introduces the notions of limits and defines infinitesimals in terms of limits. An infinitesimal variable is considered to be a sequence whose limit is zero. Second Lesson: Definition of continuity f(x+i)-f(x) is infinitesimal Third Lesson: Definition of derivative: Twenty-First Lesson: Definition of integration –Partition [x 0,X] into [x 0,x 1 ], …,[x n-1,X] –Sum: S= (x 1 -x 0 )f(x 0 )+(x 2 -x 1 )f(x 1 )+ … + (X-x n-1 )f(x n-1 ) –Take the limit with more and more intermediate values. Fixing  x=h=dx rewrite S=  h f(x)=  f(x)  x which becomes in the limit. The notation for the bounds is due to Fourier. The additivity propriety of the integral with respect to the domain is also given.

Augustin Louis Cauchy ( ) Lectures on the Infinitesimal Calculus. Twenty-Sixth Lesson: Indefinite integrals are defined and using the Intermediate Value Theorem for Integrals, it is shown that F (x) is continuous. Moreover F (x) is differentiable and F (x)’(x)=f(x). This is a version of the Fundamental Theorem. Applications: –Q.: Solve  ’(x)=0! A.:  (x)=c. –Q.: Solve y’=f(x)! A.: with  ’(x)=0 or y=∫f(x)dx=F(x)+  (x), where F(x) a particular solution. –Set F (x)= then F (x)=F(X)-F(x 0 ) for any particular solution F of F’(x)=f(x)

Further Developments Bernhard Riemann ( ) improved Cauchy’s definition by using the sums S= (x 1 -x 0 )f(c 0 )+(x 2 -x 1 )f(c 1 )+ … + (X-x n-1 )f(c n-1 ) with x i ≤c i ≤x i+1. which are now called Riemann sums. With this definition it is possible to integrate more functions. Henri Léon Lebesgue ( ) found a new way to define integrals, with which it is possible to integrate even more functions. For this one uses so- called simple functions as an approximation and measures their contribution by what is called a Lebesgue mesure. This is technically more difficult and outside the scope of usual calculus classes. It is however the integral of choice and is used e.g. in quantum mechanics. The Lebesgue integral can for instance be used to integrate the function f(x) defined by Dirichlet which is given by f(x)=1 if x is irrational and f(x)=0 if x is rational. The answer is 1. Notice that the limit of the Riemann sums does not exist, however.