Augustin Louis Cauchy (1789-1857) was a French mathematician who made significant contributions to analysis. He defined limits, continuity, derivatives, and integrals rigorously using infinitesimals. In his 1821 text, Cours d'analyse, he developed the basic theorems of calculus rigorously. His lectures on infinitesimal calculus defined integrals as the limit of Riemann sums, introducing modern notation. Later, Bernhard Riemann and Henri Lebesgue improved upon Cauchy's definition of the integral to define more functions.

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2. Augustin Louis Cauchy (1789-Augustin Louis Cauchy (1789-
1857)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,

3. Augustin Louis Cauchy (1789-Augustin Louis Cauchy (1789-
1857)1857)
Lectures on the InfinitesimalLectures on the Infinitesimal
Calculus.Calculus.
i
xfixf
y
x )()( −+
=
∆
∆
 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 [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 ∆x=h=dx rewrite S=Σ h f(x)=Σ 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.
∫
X
x
dxxf
0
)(

4. Augustin Louis Cauchy (1789-Augustin Louis Cauchy (1789-
1857)1857)
Lectures on the InfinitesimalLectures on the Infinitesimal
Calculus.Calculus.
∫=
X
x
dxxf
0
)(:
 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 ω’(x)=0! A.: ω(x)=c.
◦ Q.: Solve y’=f(x)! A.:
with ω’(x)=0 or y=∫f(x)dx=F(x)+ω(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)
∫ +=
x
x
xdxxfy
0
)()( ω
∫
X
x
dxxf
0
)(

5. Further DevelopmentsFurther Developments
 Bernhard Riemann (1826 -1866) improved Cauchy’s definition by
using the sums
S= (x1-x0)f(c0)+(x2-x1)f(c1)+ … + (X-xn-1)f(cn-1)
with xi≤ci≤xi+1.
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.

6. Further DevelopmentsFurther Developments
 Bernhard Riemann (1826 -1866) improved Cauchy’s definition by
using the sums
S= (x1-x0)f(c0)+(x2-x1)f(c1)+ … + (X-xn-1)f(cn-1)
with xi≤ci≤xi+1.
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.