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fourier_series

fourier_series - Fourier Series Fourier series started life...

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Fourier Series Fourier series started life as a method to solve problems about the flow of heat through ordinary materials. It has grown so far that if you search our library’s catalog for the keyword “Fourier” you will find 618 entries as of this date. It is a tool in abstract analysis and electromagnetism and statistics and radio communication and . . . . People have even tried to use it to analyze the stock market. (It didn’t help.) The representation of musical sounds as sums of waves of various frequencies is an audible example. It provides an indispensible tool in solving partial differential equations, and a later chapter will show some of these tools at work. 5.1 Examples The power series or Taylor series is based on the idea that you can write a general function as an infinite series of powers. The idea of Fourier series is that you can write a function as an infinite series of sines and cosines. You can also use functions other than trigonometric ones, but I’ll leave that generalization aside for now, except to say that Legendre polynomials are an important example of functions used for such more general expansions. An example: On the interval 0 < x < L the function x 2 varies from 0 to L 2 . It can be written as the series of cosines x 2 = L 2 3 + 4 L 2 π 2 X 1 ( - 1) n n 2 cos nπx L = L 2 3 - 4 L 2 π 2 cos πx L - 1 4 cos 2 πx L + 1 9 cos 3 πx L - · · · (5 . 1) To see if this is even plausible, examine successive partial sums of the series, taking one term, then two terms, etc. Sketch the graphs of these partial sums to see if they start to look like the function they are supposed to represent (left graph). The graphs of the series, using terms up to n = 5 do pretty well at representing the parabola. 0 1 3 5 x 2 1 3 5 x 2 The same function can be written in terms of sines with another series: x 2 = 2 L 2 π X 1 ( - 1) n +1 n - 2 π 2 n 3 ( 1 - ( - 1) n ) ) sin nπx L (5 . 2) and again you can see how the series behaves by taking one to several terms of the series. (right graph) The graphs show the parabola y = x 2 and partial sums of the two series with terms up to n = 1 , 3 , 5 . James Nearing, University of Miami 1

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5—Fourier Series 2 The second form doesn’t work as well as the first one, and there’s a reason for that. The sine functions all go to zero at x = L and x 2 doesn’t, making it hard for the sum of sines to approximate the desired function. They can do it, but it takes a lot more terms in the series to get a satisfactory result. The series Eq. ( 5.1 ) has terms that go to zero as 1 /n 2 , while the terms in the series Eq. ( 5.2 ) go to zero only as 1 /n .* 5.2 Computing Fourier Series How do you determine the details of these series starting from the original function? For the Taylor series, the trick was to assume a series to be an infinitely long polynomial and then to evaluate it (and its successive derivatives) at a point. You require that all of these values match those of the desired function at that one point. That method won’t work in this case. (Actually I’ve read that it can work here too, but with a ridiculous amount of labor and some mathematically suspect procedures.)
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fourier_series - Fourier Series Fourier series started life...

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