Find a few terms of the Fourier series for the given functions, and sketch at le

Question

Find a few terms of the Fourier series for the given functions, and sketch at least three periods of the function. 1. f(x) ={ 1 -pi <= x < 0 , 0 0<= x < pi 2. f(x) ={ -1 -pi <= x < 0 , 1 0<= x < pi 3. f(x) ={ 1 -pi <= x < 0 , 2 0 <= x < pi 4. f(x) ={ 0 -pi <= x < 0, pi/2<= x < pi 5. f(x) ={ 0 -pi <= x < 0 , x 0<= x < pi

Explanation / Answer

Basic Definitions A function f (x) is said to have period P if f (x + P) = f (x) for all x. Let the function f (x) has period 2p. In this case, it is enough to consider behavior of the function on the interval [-p, p]. examples Let the function f (x) be 2p-periodic and suppose that it is presented by the Fourier series: Calculate the coefficients a0, an, and bn. Solution. To define an, we integrate the Fourier series on the interval [-p, p]: For all n > 0, Therefore, all the terms on the right of the summation sign are zero, so we obtain In order to find the coefficients an at m > 0, we multiply both sides of the Fourier series by cos mx and integrate term by term: The first term on the right side is zero. Then, using the well-known trigonometric identities, we have if m ? n. In case when m = n, we can write: Thus, Similarly, multiplying the Fourier series by sin mx and integrating term by term, we obtain the expression for bm: Rewriting the formulas for an, bn, we can write the final expressions for the Fourier coefficients: Example 2 Find the Fourier series for the square 2p-periodic wave defined on the interval [-p, p]: Solution. First we calculate the constant a0: Find now the Fourier coefficients for n ? 0: Since , we can write: Thus, the Fourier series for the square wave is We can easily find the first few terms of the series. By setting, for example, n = 5, we get The graph of the function and the Fourier series expansion for n = 10

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