Augustin Louis Cauchy (1789-1857)

1
Augustin Louis Cauchy (1789-1857)

• 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.
• 1817 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,

2
Augustin Louis Cauchy (1789-1857)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(xi)-f(x) is infinitesimal
• Third Lesson Definition of derivative
• Twenty-First Lesson Definition of integration
• Partition x0,X into x0,x1, ,xn-1,X
• Sum S (x1-x0)f(x0)(x2-x1)f(x1)
  (X-xn-1)f(xn-1)
• Take the limit with more and more intermediate
  values.
• Fixing Dxhdx rewrite SS h f(x)S f(x) Dx
  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.

3
Augustin Louis Cauchy (1789-1857)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 w(x)0! A. w(x)c.
• Q. Solve yf(x)! A.
• with w(x)0 or y?f(x)dxF(x)w(x), where
  F(x) a particular solution.
• Set F(x) then F(x)F(X)-F(x0)
• for any particular solution F of F(x)f(x)

4
Further Developments

• Bernhard Riemann (1826 -1866) improved Cauchys
  definition by using the sums
• S (x1-x0)f(c0)(x2-x1)f(c1) (X-xn-1)f(cn-1)
• with xicixi1.
• which are now called Riemann sums. With this
  definition it is possible to integrate more
  functions.
• Henri Léon Lebesgue (1875-1941) 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.