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When analyzing situations as diverse as electrical oscillations, vibrating mechanical systems, longitudinal oscillations in crystals, and many other physical phenomena, Fourier series are found to arise naturally. Furthermore, the individual terms in a Fourier series often have an important physical interpretation.

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When analyzing situations as diverse as electrical oscillations, vibrating mechanical systems, longitudinal oscillations in crystals, and many other physical phenomena, Fourier series are found to arise naturally. Furthermore, the individual terms in a Fourier series often have an important physical interpretation. In a vibrating mechanical system, for example, each component of a Fourier series representation of the overall vibration represents a fundamental mode of vibration. The full Fourier series shows how each mode contributes to the solution, and which the most significant modes are. This information can often be used to advantage, either by showing how the modes can be utilized to achieve a desired effect, or by using the information to enable systems to be constructed that minimize undesirable vibrations. It is for these and other reasons that it is necessary for engineers and physicists to study the properties of Fourier series.A Fourier series representation of a function f(x) over the interval is an expression of the form (1)where the coefficients are determined by the function . It is important to notice that the Fourier series representation of contains two infinite sums, one of even functions (the cosines) and the other of odd functions (the sines). It would be recalled that a function defined in the interval is said to be an even function in the interval if (2)and to be an odd function in the interval if .(3)The cosine fun

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