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0.9r == 1.0r
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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.
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