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Fourierreihen und Fouriertransformation vi chaeology, medicine, agriculture, natural history and so on were being carried out at a time when Napoleon was fighting Syrians in Palestine, repelling Turkish invasions, hunting Murad Bey, the elusive Mameluke chief, and all this without support of his fleet, which had been obliterated by Nelson at the Battle of the Nile immediately after the disembarkation. Shortly before the military capitulation in 1801, the French scientists put to sea but were promptly captured with all their records by Sidney Smith, commander of the British fleet. However, in accordance with the gentlemanly spirit of those days, Smith put the men ashore, retained the documents and collections for safekeeping, and ultimately delivered the material to Paris in person, except for the Rosetta stone, key to Egyptian hieroglyphics, which stands today in the British Museum memorializing both Napoleon's launching of Egyptology and his military failure. The English physicist Thomas Young (1773-1829), father of linearity, is well known for establishing the transverse wave nature of light, explaining polarization, and also for introducing the doublt/pinhole interferometer, which exhibits the Fourier analysis of an optical object. Less well known is that he shared an interest in Egyptology with Fourier: he worked on the Rosetta stone, explained the demotic and hieratic scripts as descended from hieroglyphic writing, and isolated and identified consonantal signs. Fourier was appointed as Prefect of Isere by Napoleon in 1802 after a brief return to his former position as Professor of Analysis at the Ecole Polytechnique in Paris. His duties in Grenoble included taxation, military recruiting, enforcing laws, and carrying out instructions from Paris and writing reports. He soothed the wounds remaining from the Revolution of 1789, drained 80 000 km 2 of malarial swamps, and built the French section of the road to Torino. By 1807, despite official duties, Fourier had written down his theory of heat conduction, which depended on the essential idea of analyzing the temperature distribution into spatially sinusoidal components; but doubts expressed by Laplace and Lagrange hindered publication. Criticisms were also made by Biot and Poisson. Even so, the Institut set the propagation of heat in solid bodies as the topic for the prize in mathematics for 1811, and the prize was granted to Fourier but with a citation mentioning lack of generality and rigor. The fact that publication was then further delayed until 1815 can be seen as an indication of the deep uneasiness about Fourier analysis that was felt by the great mathematicians of the day. It is true that the one-dimensional distribution of heat in a straight bar would require a Fourier integral for its correct expression. Fourier avoided this complication by considering heat flow in a ring, that is, a bar that has been bent into a circle. In this way, the temperature distribution is forced to be spatially periodic. There is essentially no loss of generality because the circumference of the ring can be supposed larger than the greatest distance that could be of physical interest on a straight bar conducting heat. This idea of Fourier remains familiar as one of the textbook methods of approaching the Fourier integral as a limit, starting from a Fourier series representation. Fourier was placed in a tricky position in 1814, when Napoleon abdicated and set out for Elba with every likelihood of passing southward through Grenoble, on what has come to be known today as the Route Napoleon. To greet his old master would jeopardize his standing with the new king, Louis XVIII, who in any case might not look favorably on old associates and appointees of the departing emperor. Fourier influenced the choice of a changed route and kept his job. But the next year Napoleon reappeared in France, this time marching north through Grenoble where he fired Fourier, who had made himself scarce. Nevertheless, 3 days later Fourier was appointed Prefect of the Rhone at Lyons, thus surviving two changes of regime. Of course, only 100 days elapsed before the king was back in control and Napoleon
Fourierreihen und Fouriertransformation vii was on his way to the south Atlantic, never to return. Fourier's days in provincial government then ended and he moved to Paris to enter a life of science and scientific administration, being elected to the Academie des Sciences in 1817, to the position of permanent secretary in 1823, and to the Academie Française in 1826. He never married. At the beginning mention was made of Euler's formula. The formula is correct for π < x < π but not for other ranges of x. The right hand side is the Fourier series for the sawtooth periodic function ⎛ x ⎞ ⎞ ⎛ ⎟ ⎞ ⎜ ⎛ x ⎟ ∗ 1 x ⎜ ⎟Π ΙΙΙ ⎜ . ⎝ 2 ⎠ ⎝ 2π ⎠ 2π ⎝ 2π ⎠ Fourier wrote, around 1808-1809: “the equation is no longer true when the value of x is between π and 2π. However, the second side of the equation is still a convergent series but the sum is not equal to x/2. Euler, who knew this equation, gave it without comment.” (Quotation from J. Herivel, Joseph Fourier, the Man and the Physicist, Clarendon Press, Oxford, 1975, p. 319.)
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