Question on the history of math - Fourier

[quasar987]

It is often said in textbook how the paper by Fourier introducing Fourier series was submitted to the academy of science but rejected or not taken seriously.

How could that have been, since his idea is so clear and simple?

"Suppose a function is integrable and can be written as a trigonometric series, then the coefficients are given by this integral: [...]"

 

[JasonRox]

I'm not surprised.

Great papers are usually rejected (and probably stolen later).

 

[yasiru89]

Somewhat political really. Fourier was a political critic and the panel including Lagrange saw the lack of rigour and made that reason to criticize and reject the paper.
Fourier didn't investigate conditions for validity and made use of the results as a means to an end for his heat conduction exposition and it is true that evidence points that Euler had foreseen this development early on but the result is still impressive.
It didn't really matter since some of the panel were impressed and it took Dirichlet, another Frenchman to put in the rigour and populate it.

 

[trambolin]

How can one be rigorous to Lagrange after reading analytical mechanics?

 

[HallsofIvy]

First Fourier was an engineer, not a mathematician (my understanding is that he was chief of engineers in Napolean's expedition to Egypt) so that may have had an influence.

However, those who rejected his paper were quite right: what he said was not valid.

Fourier made two claims: that any "square integrable" function had a Fourier, sine and cosine, series and that any such series, with some conditions on the coefficients, corresponded to such a function. The first is obviously true, just by doing the integration to exhibit the coefficients. The second is false- there existed such series that did NOT converge to such functions. In order for Fourier's method to be valid, both statements had to be true.

Yet, engineers went ahead blithely using Fourier's method to solve differential equations, getting solutions that were clearly correct. That was a major reason for developing the "Lesbesque Integral" which was unknown up to that time. Both statements ARE true if you use the Lebesque Integral rather than the Riemann Integral.

 

[Gokul43201]

First Fourier was an engineer, not a mathematician (my understanding is that he was chief of engineers in Napolean's expedition to Egypt) so that may have had an influence.

When Fourier returned from Egypt, he brought back some ancient glyphs and artifacts which he showed a young family friend (or maybe even a nephew of some kind) by the name of Jean Francois Champollion. Champollion was something of a linguistic prodigy - not yet a teenager, he could speak over a dozen different languages. Under Fourier's influence, Champollion became excited about Egyptology and particularly about understanding the ancient writing. A few years later, Champollion (who, thanks to some string-pulling from Fourier, was able to avoid conscription) solved a centuries old problem by completely deciphering the Egyptian hieroglyphics.

However, those who rejected his paper were quite right: what he said was not valid.

Fourier made two claims: that any "square integrable" function had a Fourier, sine and cosine, series and that any such series, with some conditions on the coefficients, corresponded to such a function. The first is obviously true, just by doing the integration to exhibit the coefficients. The second is false- there existed such series that did NOT converge to such functions.

The proof of the converse was, in fact, a pretty old problem that, among others, Langrange himself had wrestled with, to very limited success. It was therefore unacceptable to him that Fourier would simply assert its truth and proceed with the calculation of the heat transfer problem. I think it was over a decade later that Dirichlet provided the final word on the matter that the converse was if fact, not always true.

In the (translated) words of Carl Jacobi,"Only Dirichlet, not I, not Cauchy, nor Gauss, knows what a perfectly rigorous proof is, but we learn it only from him. When Gauss says he has proved something, I think it is very likely; when Cauchy says it, it is a fifty-fifty bet; when Dirichlet says it, it is certain."