11.1-11.6과제 풀이

11.1-11.6과제 풀이

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Unformatted text preview: Homework 1 § 11 . 1 1. If f ( x ) has period p , show that f ( ax ), a 6 = 0, is a periodic function of x of period p/a . Sol. Let g ( x ) = f ( ax ). Since p is a period of f , g ( x + p/a ) = f ( a ( x + p/a )) = f ( ax + p ) = f ( ax ) = g ( x ) . Thus g ( x ) = f ( ax ) is a periodic function of x of period p/a . 2. Find the Fourier series of the given periodic function f ( x ) of period 2 π : (a) (b) (c) f ( x ) = x 2 (- π < x < π ) Sol. (a) a = 1 2 π Z π- π f ( x ) dx = 1 π Z π/ 2 dx = 1 2 , a n = 1 π Z π- π f ( x ) cos nxdx = 2 π Z π/ 2 cos nxdx = 2 sin nπ 2 nπ , b n = 1 π Z π- π f ( x ) sin nxdx = 1 π Z π/ 2- π/ 2 sin nxdx = 0 . Fourier series of f ( x ) f ( x ) = 1 2 + ∞ X n =1 2 sin nπ 2 nπ cos nx = 1 2 + 2 π £ cos x- 1 3 cos 3 x + 1 5 cos 5 x + ··· / . Computational Science & Engineering (CSE) C. K. Ko Homework 2 (b) a = 1 2 π Z π- π f ( x ) dx = 1 π Z π (- x + π ) dx = π 2 , a n = 1 π Z π- π f ( x ) cos nxdx = 2 π Z π (- x + π ) cos nxdx = 2(1- cos nπ ) n 2 π , b n = 1 π Z π- π f ( x ) sin nxdx = 0 . Fourier series of f ( x ) f ( x ) = π 2 + ∞ X n =1 2(1- cos nπ ) n 2 π cos nx = π 2 + 4 π £ cos x + 1 3 2 cos 3 x + 1 5 2 cos 5 x + ··· / . (c) a = 1 2 π Z π- π f ( x ) dx = 1 π Z π x 2 dx = π 2 3 , a n = 1 π Z π- π f ( x ) cos nxdx = 2 π Z π x 2 cos nxdx = 4 cos nπ n 2 , b n = 1 π Z π- π f ( x ) sin nxdx = 1 π Z π- π x 2 sin nxdx = 0 . Fourier series of f ( x ) f ( x ) = π 2 3 + ∞ X n =1 4 cos nπ n 2 cos nx = π 2 3- 4 £ cos x- 1 2 2 cos 2 x + 1 3 2 cos 3 x + ··· / . 3. Using Problem 2, find the sum of the series. (a) ∞ X n =1 (- 1) n- 1 1 2 n- 1 (b) ∞ X n =1 1 (2 n- 1) 2 (c) ∞ X n =1 (- 1) n- 1 1 n 2 Sol. (a) Notice that f is continuous at x = 0. 1 = f (0) = 1 2 + 2 π £ 1- 1 3 + 1 5 + ··· / Thus ∞ X n =1 (- 1) n- 1 1 2 n- 1 = π 4 . Computational Science & Engineering (CSE) C. K. Ko Homework 3 (b) Notice that f is continuous at x = 0....
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This note was uploaded on 11/08/2011 for the course EE 111 taught by Professor Kim during the Spring '11 term at Korea University.

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11.1-11.6과제 풀이

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