The Origin Of The Theory Of Fourier Series

The theory of Fourier Series with a history of almost two hundred years has now developed into one of the the core explored fields of modern mathematics. On the one hand, it is tightly connected with many branchs of mathematics such as the theory of Partial Differential Equations, Complex Variable Function, Probability, Algebra, Topology and so on. On the other hand, it is the main means for solving problems in Engineering Technology, Classical Physics and Quantum Mechanics.It is used extensively in thermotics, optics, electromagnetism, aerodynamics, bionics, biology and so on. The discovery of Fourier Series is considered as a great event in history of mathematics, since it ended completely the argument on the problem of vibrating string and leaded mathematical analysis being rigor. Although the origin of the theory of Fourier Series had been explored by some scholars from abroad, there are many problems, needing to discussing further.Historical research method, literature method and comparison method are mainly adopted in this study to explore the origin of the theory of Fourier Series. The major contributions are as follows:1.The origin of the theory of Fourier Series is explored from varied aspects like music, physics,Mathematics and the tendency of science.It is found that "the theory of Fourier Series was based on the notion of superposition of simple modes"2. The reason why Taylor and Johann Bernoulli had not found the equation of vibrating string and the solution of higher modes is explored generally. As is shown that, on the one hand, Taylor and Johann Bernoulli did not pay close attention to the shape of vibrating string but were interested in the time of oscillation instead of oscillation itself. On the other hand, because of lacking of calculus of trigonometric function, Taylor and Johann Bernoulli had to integrate by geometrical means, which prevented them from putting forward the solution of higher modes.3. The reason why Fourier had gone in for the study of communication of heat is investigated. According to the investigation, the promotion of mathematization of science, the influence of great change of experimental physics in Napoleonic France, the establishment of thermometry and calorimetry were the major factors promoting Fourier to study communication of heat.4. The influence of Biot's paper in 1804 on Fourier's study about communication of heat is explored in accordance with the analysis of the content and structure of Fourier's Draft Paper, the paper in 1807, Award-winning paper and The Analytical Theory of Heat. The result indicates that both the popular viewpoint "Biot's paper in 1804 enlightened Fourier to study communication of heat in continuous object instead of between discrete bodies" and Olivier Darrigol's viewpoint "Biot's paper in 1804 enlightened Fourier to go into the field of communication of heat between discrete bodies" are dubious, for these viewpoints were short of persuasive evidence.But it is certain that the difficulty of inhomogeneous exerted great influence on Fourier's study on communication of heat.The concept of heat flux was the outcome of Fourier overcoming the difficulty of inhomogeneous, while the accurate expression of heat flux was the key for Fourier to setting up the equation of communication of heat in the interior of solid body.5. The reason why Fourier could set up his theory of trigonometric series successfully is also discussed. The discussion reflects that the enlightenment of the notion of superposition of simple modes, the inspiration of having solved the problem of communication of heat in semi-infinite strip and between discrete bodies arranged around semi-perimeter of a circle, the reasonable understanding of the concept of function, the demand for solving equation of heat conduction were important factors prompting Fourier to finish his theory of trigonometric series successfully.6. The influence of the theory of Fourier Series on theoretical physics(including applied mathematics) and pure mathematics was examined. It was found that the theory of Fourier Series produced a great impact on the masters of Cambridge Mathematical Physics School such as George Green,George Gabriel stokes, Joseph John Thomson, James Clark Maxwell and so on.The impact on Joseph John Thomson seems much greater, whose whole career of mathematical physics was connected with the theory of Fourier Series. Moreover, the theory of Fourier Series also exerted a great influence on pure mathematics.The formation of uniform convergence concept, the establishment of Riemann integral and the modern function concept, separating differentiable function from continuous function, the construction of set theory by Georg Cantor were closely connected with the theory of Fourier Series.