Jean Baptiste Joseph Fouriers Contributions to Mathematics - Coursework Example

Running Head: Fourier’s Contribution Jean Baptiste Joseph Fourier’s Contributions to Mathematics of Jean Baptiste Joseph Fourier was a French mathematician known chiefly for his contribution to the mathematical analysis of heat flow. Throughout his life Fourier pursued his interest in mathematics and mathematical physics. This paper intends to present a detailed account of his development as a mathematician and his contribution in the field of Mathematics. The paper begins with his biographical introduction and proceeds towards his achievements with specific reference to “Fourier Analysis” and “Fourier Series”. An effort has been made to present complete historical background of his contributions. Jean Baptiste Joseph Fourier’s Contributions to Mathematics Introduction Fourier, Jean Baptiste Joseph was born in 1768. He was a French mathematician and administrator, who is famous as the creator of Fourier series. Fourier was born in Auxerre on March 21, 1768, the 19th child of a tailor. Orphaned at eight, he was educated in the Benedictine school at Auxerre, where he discovered his interest in mathematics. His thoughts of becoming a monk were ended by the French Revolution: he was a student at the Ecole Normale in Paris in 1794 and became assistant professor at the founding of the Ecole Polytechnique in 1795. In 1798, Fourier accompanied Napoleon on his campaign in Egypt, where in addition to many administrative appointments he also served as secretary of the Institut dEgypte. He suggested the preparation of a record of French discoveries in Egypt, to which he made noteworthy contributions; it appeared as the Description de lEgypte between 1809 and 1828. After his return to France he was made Prefect of Isere in 1802. He served with great distinction for 13 years, being elected to a barony in 1808. In 1815, Napoleon, returning from Elba, made Fourier Prefect of the Rhône department, but Fourier resigned before the end of the Hundred Days in protest against Napoleons regime. Fourier spent the rest of his life in Paris. His Napoleonic past made it difficult for him to find employment, but in 1817 he was elected to the Academie des Sciences, and in 1822 he became perpetual secretary for mathematical sciences. He died in Paris on May 16, 1830. (Cantor and Jourdain, 1955) Contribution to Mathematics Fouriers most important achievements were concerned with heat diffusion and with the mathematical techniques that he developed in its study. He introduced the "diffusion equation" to express the movement of heat within a body, and then the important novelty of a separate equation to deal with heat movement on the surface. In solving the diffusion equation, he discovered, as a means of relating the full solution to the boundary values (the temperatures of the surfaces of the body) the Fourier series representation of a mathematical function. Fourier presented his results to the Academie des Sciences in a long paper in December 1807. The controversial nature of Fourier series led to its rejection by Joseph Louis Lagrange, who had in his own work rejected such series. However, a prize problem in the subject was set for 1812, and Fourier won the prize with a new edition of the paper which included also the "Fourier integral" for heat diffusion in infinite bodies. Again publication was delayed; so Fourier prepared a third version as La theorie analytique de la chaleur (1822; Eng. tr., The Analytical Theory of Heat, 1878). His work, especially Fourier series, became immensely influential in mathematical physics, theoretical astronomy, and engineering, and it had great impact in pure mathematics, where it led to revised formulations of functions, series, and integrals. In addition to further studies of heat, Fourier spent his last years on a work reflecting his lifelong interest in the theory of equations. His most important results were the inductive proof (which he found in his teens) of Descartes rule of signs, and others anticipating the modern interest in linear programming. Only some parts of this work were fit for posthumous publication, as Analyse des equations determinees (1831). (Herivel, 1975) Fourier’s Analysis Fourier analysis is a branch of mathematics that is used to analyze repeating, or periodic, phenomena. Many natural and artificial phenomena occur in cycles that repeat constantly. These phenomena such as alternating currents, business cycles, high and low tides, the orbits of planets and artificial satellites, and the vibrations of electromagnetic waves can be described by a mathematical concept called a function. Since these phenomena are periodic, their functions are called periodic functions. In general, a function is said to be periodic if its graph is a repeating pattern. The basic goal of Fourier analysis is to represent periodic functions in terms of series of particular, and generally simpler, periodic functions. Most of the simpler functions occur in trigonometry and are therefore called trigonometric functions. They are often used in Fourier analysis to expand a given function. If the function does possess a Fourier series, the coefficients can be calculated by means of integral calculus. Fourier analysis was first developed by Joseph Fourier in the 1820s and has been highly elaborated. An analogous method making use of a small mathematical fluctuation called a "wavelet" was developed in the late 1980s. Wavelet analysis offer advantages in analyzing rapidly changing signals and in handling gaps in a body of data. (Folland 1992) Fourier Analysis Adopted from Cochlear implants Fourier’s Series Fourier series, a mathematical expression that is important both in mathematics itself and in a wide variety of applications in the physical sciences, especially in the theory of wave motion. The series is named after him as he discovered the series in 1807. Fourier Series : Adopted from Fourier Series We can most easily introduce the series by first using an example in which we consider how a composite musical tone can be represented as a sum of pure tones. The graph of a composite tone is shown in the accompanying diagram, top, where A is the amplitude of the tone, T is the period, and t is the time. The graph of a pure tone, represented by a sine curve, is shown in the diagram, bottom. We notice that the curve shown in the diagram above, top has a pattern that repeats after the period T. Thus, the function f, which represents this curve, is a periodic function with a period T; that is, f (t + T) = f (t). The sine curve shown in the diagram above, bottom also is a periodic function with period T. The key idea for the uses of a Fourier series is that any periodic function F can be represented by summing a series of trigonometric terms which are called the expansion of the function F. For the representation of the function f shown in the previous diagram, top, we form the sine terms: (1) Sin (2t/T) + sin (22t/T) + sin (23t/T) + sin (2kt/T), Where the first term represents the fundamental tone and the succeeding terms represent the harmonics of the fundamental tone. Such a procedure gives a harmonic analysis of the function. A sine function has a value of zero at t = 0, and it is an odd function; that is, sin (–t) = –sin t. In order to have an expression with full generality we must add a series of cosine terms to the sine terms given in (1). A cosine function, such as that shown in the previous diagram, bottom, is an even function; that is, cos (–t) = cos t. By using cosine terms and sine terms, we obtain the full harmonic analysis of the composite tone shown in the previous diagram, top: (2) Where denotes summation. It may seem strange that a sum of smooth cosine and sine curves can reproduce every little bump in the curve shown in that diagram, but it is so. When only several terms of a series are used, we obtain an approximation to the shape of a curve. An example of such an approximation is shown in the illustration below. The formulas for evaluating the coefficients ak and bk in Equation (2) were first established by Fourier. These formulas, which were one of his great achievements, are: (3) The Fourier series expansion of any periodic function F is then written out as: (4) In Equation (4), which was obtained by putting the coefficients given by Equation (3) into their place in the right side of Equation (2), we have set T = 2 for simplicity of notation. (Georgi et. al, 1976) Historical Background As stated by Button and Wiltse (1979) the possibility of forming a trigonometric expansion of a periodic function had been recognized by predecessors of Fourier. For instance, about 1750 Daniel Bernoulli used a series of trigonometric terms in his proposed solution for representing the motion of a vibrating string. He advocated this procedure as a general principle, but he did not calculate the coefficients of the terms. Fourier was the first one to recognize the significance of the coefficients. He saw that the trigonometric expansion could be applied independently of whether a function f(x) is periodic. He also saw that a function need not be smooth to have a trigonometric expansion but could have steep sides like those of a square wave. Fourier developed the use of his series into a powerful general method for solving problems such as those involved in heat diffusion. The key to his method is the distinction between the interior of a region and its boundary in space and in time. Fouriers method of obtaining series solutions to boundary-value problems in the partial differential equations of physics still is a basic tool in physics and engineering. In addition, the mathematical problems presented by the Fourier series have been a great stimulus to the development of mathematics. References Button, K. J. & Wiltse, J. C. (1979). Infrared and Millimeter Waves: Instrumentation, Vol. II,Academic Press, ISBN 0121477029, 9780121477028, pp.139-152 Cantor, G. & Jourdain, P. E. (1955).Contributions to the Founding of the Theory of Transfinite Numbers, Courier Dover Publications, ISBN 0486600459, 9780486600451, pp. 1-4 Cochlear implants – wiring for sound, Nova Science in the News, Published by the Australian Academy of Science, retrieved June 8, 2009 from: http://www.science.org.au/nova/029/029print.htm Folland, G. B. (1992). Fourier Analysis and its Applications, Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 0534170943, 9780534170943 Fourier series, Wolfram Math World, Retrieved June 8. 2009 from: http://mathworld.wolfram.com/FourierSeries.html Georgi, P., Tolstov, R., & Silverman, A. (1976). Fourier Series, Courier Dover Publications, ISBN 0486633179, 9780486633176 Herivel, J. (1975). Joseph Fourier: The Man and the Physicist, Clarendon Press, pp. 217-220 Read More