The following is from The Fourier Transform and Its Application, by Ronald N. Bracewell, McGraw-Hill, 1986,pp 462-464.
Baron Jean-Baptiste-Joseph Fourier (March 21 1768-May 16, 1830), born in poor circumstances in Auxerre, introduced the idea that an arbitrary function, even one defined by different analytic expressions in adjacent segments of its range (such as a staircase waveform), could nevertheless be represented by a single analytic expression. This idea encountered resistance at the time I but has proved to be central to many later developments in mathematics science, and engineering. It is at the heart of the electrical engineering curriculum today. Fourier came upon his idea in connection with the problem of the flow of heat in solid bodies, including the earth.
The formula
x/2 = sin x - (sin 2x)/2 + (sin 3x)/3 + · · ·
was published by Leonhard Euler (1707-1783) before Fourier’s work began, so you might like to ponder the question why Euler did not receive the credit for Fourier’s series.
Fourier was obsessed with heat. keeping his rooms uncomfortably hot for visitors, while also wearing a heavy coat himself Some traced this eccentricity back to his 3 years in Egypt where he sent in 1798 with the 165 savants on Napolean's expedition to civilize the country.
Prior to the expedition Fourier was a simple professor of mathematics but he now assumed administrative duties as secretary of the Institut d'Égypte a scientific body that met in the harem of the palace of the Beys Fourier worked on the theory of equations at this time but his competence at administration led to political and diplomatic assignments that h e also discharged with success. It should be recalled that the ambitious studies in geography archaeology. 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 double 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 Isère 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 km2 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 Riot and Poisson. Even so, the Institut set the propagation of heat in solid bodies as the topic for the prize in mathematics for 181 1, and the prize was granted to Fourier but with a citation mentioning lack of generality and rigor. The fact that publication w-as then further delayed until 1815 can be seen as an indication of the deep uneasiness about Fourier analysis that was fell 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. Rut 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 Rhône at Lyons, thus surviving two changes of régime Of course only 100 days elapsed before the king was back in control and Napoleon 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 Académie des Sciences in 1817, to the position of permanent secretary in 1823, and to the Académie Française in 1826. He never married.