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Fourier serial and Their Applications Rui Niu May 12, 2006 Abstract Fourier serial be of roughhewn importance in both theoretical and ap& timid; plied mathematics. For orthonormal families of illogical& incertain; cherished functions {φn }, Fourier Series be magnetic core totals of the φn that can approximate periodic, entangled& jump;valued functions with dictatorial precision. This paper will focus on the Fourier Series of the mingled exponentials. Of the many possi& startle; ble methods of estimating complex­valued functions, Fourier serial publication are specially attractive because uniform convergence of the Fourier serial (as more terms are added) is guaranteed for continuous, bounded functions. Furthermore, the Fourier coefficients are penetrative to minimize the square of the error from the actual function. Finally, complex exponentials are relatively simple to deal with and ubiquitous in individual(prenominal) phenomena. This paper  4257;rst defines generalized Fourier series, with an emphasis on the se­ ries with complex exponentials. Then, important properties of Fourier series are described and proved, and their relevance is explained. A com­ plete example is then given, and the paper concludes by briefly mentioning some of the applications of Fourier series and the generalization of Fourier series, Fourier transforms.

1 origination and priming coat Information In the mid­eighteenth century, physical problems such(prenominal) as the conduction pat­ terns of heat and the examine of vibrations and oscill ations led to the study of Fourier series. O! f central interest group was the problem of how commanding legitimate­valued functions could be represented by sums of simpler functions. As we shall see later, a Fourier series is an infinite sum of trigonometric functions that can be apply to model real­valued, periodic functions. We shall begin by giving a brief description of the trigonometric polynomials, and especially of their relation to the complex exponentials. Let us define:...If you regard to get a full essay, order it on our website: BestEssayCheap.com

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