Through Fourier's research the fact was established that an arbitrary at first, continuous [2] and later generalized to any piecewise -smooth [3] function can be represented by a trigonometric series. The first announcement of this great discovery was made by Fourier inbefore the French Academy. The heat equation is a partial differential functoins.

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Prior to Fourier's work, no solution to the heat equation was known in the general case, although particular solutions were known if the heat history of exponential functions behaved in a simple way, in particular, if the heat source was a sine or cosine wave. These simple solutions are now sometimes called eigensolutions. Fourier's idea was to model a complicated heat source as a superposition or linear combination of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions.

This superposition or linear combination is called the Fourier series. From a modern point read more view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century.

Later, Peter Gustav Lejeune Dirichlet [5] and Bernhard Riemann [6] [7] [8] expressed Fourier's results with greater precision and formality. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving gunctions differential equations with constant coefficients, for which history of exponential functions eigensolutions are sinusoids. The Fourier series has many such applications in electrical engineeringvibration analysis, acousticsopticssignal processingimage processingquantum mechanicseconometrics[9] shell theory[10] etc.]

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