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![Page 1: Fourier Series and Integral - WordPress.com · Fourier series are the basic tool for representing periodic functions, which play an important role in applications. A function f(x)](https://reader034.vdocuments.net/reader034/viewer/2022042311/5ed95875f59b0f56f45f5114/html5/thumbnails/1.jpg)
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Fourier Series Fourier Integral
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Fourier series are the basic tool for representing periodic functions, which play an important role in applications.
A function f(x) is called a periodic function if f(x) is defined for all real x and if there is some positive number p, called a period of f(x), such that
Familiar periodic functions are the cosine and sine functions . Examples of functions that are not periodic are x, x2, x3,ex, cosh x, and In x, to mention just a few.
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If f(x) has period p, it also has the period 2p because (1) implies f(x + 2p) = f([x + p] + p) = f(x + p) = f(x), etc.; thus for any integer n = 1, 2,3, ... ,
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functions f(x) of period 2π
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a trigonometric series, that is, a series of the form
ao, a1, b1, a2, b2, ... are constants, called the coefficients of the series. We see that each term has the period 2π.
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Given by Euler formulas
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Example 1.Find the Fourier coefficients of the periodic function f(x).
Solution
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From (6a) we obtain ao = O. From (6b)
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From (6c)
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The first term on the right equals 2πao. Integration shows that all the other integrals are O. Hence division by 2π gives (6a).
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Example 1
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The left part of the figure shows this function for 2L = 4, 8, 16 as well as the non periodic function f(x ), which we obtain from fL if we let L→∞,
We now explore what happens to the Fourier coefficients of fL as L increases. Since fL is even, bn = 0 for all n. For an the Euler formulas (6), Sec. 11.2, give
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This sequence of Fourier coefficients is called the amplitude spectrum of f Lbecause |an| is the maximum amplitude of the wave ancos(nπx/L).
We see that for increasing L these amplitudes become more and more dense on the positive wn-axis, where wn = nπ/L.
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We now consider any periodic function f(x) of period 2L that can be represented by a Fourier series
and find out what happens if we let L→∞
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If we insert an and bn from the Euler formulas (6), Sec. 11.2, and denote the variable of integration by v, the Fourier series of fL(x) becomes
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Then 1/L = ∆w/π, and we may write the Fourier series in the from
This representation is valid for any fixed L, arbitrarily large, but finite.
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We now let L →∞ and assume that the resulting nonperiodic function
is absolutely integrable on the x-axis; that is, the following (finite!) limits exist:
Then 1/L →0, and the value of the first term on the right side of (1) approaches zero.
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Also ∆w = π/L → 0 and it seems plausible that the infinite series in (1) becomes an integral from 0 to ∞, which represents f(x), namely,
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Example 2
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Solution
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