The Garden of Archimedes
A Museum for Mathematics Integration and measure works of the section Giuseppe Peano, Geometrical applications of infinitesimal calculus, Torino-Firenze-Roma-Napoli, Fratelli Bocca editori, 1887.
see also
In the of 1823 Cauchy gives what is identified as the first modern definition of integral. He considers the case of continuous functions on an interval, and then expands on the case of the functions with one or a finite number of discontinuity . In an 1829 article of in the "Journal für die reine und angewandte Mathematik", dealing with the problem of the representation of functions in Fourier's series, Dirichlet raises the issue of the case of functions with a finite number of discontinuity, also giving the example of the function carrying his name. In 1854 Bernhard Riemann (1826-1866) wrote his thesis for the University teaching qualification, entitled , that is "On the possibility to represent a function through a trigonometric series", which remained unknown until 1867 when it was published by Dedekind. Here, the integral called after him is introduced and is supplied with examples of functions that, even with an infinite number of discontinuity, turn out to be integrable. Numerous researches are grafted on this new definition. They relate to the properties of the subsets of the straight line, first among them those of Cantor. More and more contributions were added about the characterisation of integrability in relation to the set of points of discontinuity like the ones from Hermann Henkel, Paul Du Bois - Reymond, Henry Smith, Axel Harnack, Vito Volterra. | |
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