II.5 Rigor in Mathematical Analysis: Cauchy, Riemann, the Integral and Counterexamples: Pages 123 – 124

Cauchy revisited the definition of the convergence of a sum of an infinite series to be the limit of the sequence of partial sums.  This unified the approach for a series of numbers and a series of functions.  This approach is called the “arithmetization of analysis”.  He defined a continuous function as one for which “an infinite  increase of the variable produces an infinitely small increase of the function itself”.

Cauchy understood the importance of jumps in the function and their relationship to understanding the integral of the function.  Cauchy did assume continuity when defining the definite integral in his lectures.

Cauchy’s ideas became standard in France and in French schools.  Abel, Dirichlet and Riemann all studied Cauchy’s ideas.

Bolzano a Bohemian priest and Weierstrass of Germany also independently had ideas related to those of Cauchy. 

Riemann generalized the ideas of Cauchy for functions that are not necessarily continuous.  Riemann defined the Riemann integral in his second thesis.  He provided an example of a function that is discontinuous on any interval but yet can still be integrated. The integral thus has points of non-differentiability on each interval.  “Riemann’s definition rendered problematic the the inverse relationship between differentiation and integration …”.  This and the use of other counterexamples flourished at this time.  This definition was published in 1867 only after Riemann’s death.

Gaston Darboux published an expository version of the Riemann integral in 1873.  The rigor used by Riemann was appreciated later by the Weierstrass school.  Cantor also later utilized the work of Cauchy in the 1870’s in the investigations of point sets.

The Dirichlet problem is: “Given a function g, defined on the boundary of a closed region in the plane, does there exist a function f that satisfies the Laplace partial differential equation in the interior and takes the same values as g on the boundary?”  Riemann asserted that the answer to this question is “yes”.  Weierstrass published a counterexample in 1870. In 1900 Hilbert expressed ideas which made more precise and broad hypothesis that rehabilitated the Dirichlet problem.

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