Question

1. True or false: (a) The constant term of the Fourier series representing f(x) 2,-2<2,f(x +4) f(z), is o 4 2 3 (b) The Fourier series (of period 2T) representing f(x)-3 - 7sin2(z) is a Fourier sine series (c) The Fourier series of f(x) = 3x2-4 cos22, -π < x < π, f(x + 2π) = f(x) is a cosine series (d) Every Fourier sine series converges to 0 at x = 0 (e) Every Fourier sine series has 0 as its constant term. (f) The constant term of the Fourier series representing f(z) = 22, -1 < x < 1, f(x + 2) = f(z), is zero.

Answer 1

Given collected from data is let us the consider from the data as followed (i) Let the constant into terms of the fouries series f(x) interval - L lessthanorequalto x lessthanorequalto L is defined as f(x) = a_0/2 + Sigma_n = 1^infinity_0 an cos(n pi x/L) + Sigma_n = 1^infinity bn sin (n pi x/L) = > a_0/2 = 1/2L integral^L_-L f(x) dx we have a-n = 1/L integral_-L^L f(x) cos(n pi x/L) dx, n > 0 b_n = 1/L integral^L-L f(x) sin (n pi x/L)dx, n > 0 (a) let the f(x) = x^2 - 2 < x < 2, f(x + 4) = f9x) is a_0/2 = 4/3 L = 2 we have a_0/2 = 1/4 integral_-2^2 x^dx = > 1/4 [x^3/3]^2-2 = > 16/4.3 = 4/3 = > a_0/2 - 4/3 is true \r\n (b) The following series representing f(x) = 3 - 7 sin^2(x) let us consider the data = > a_n = 1/pi integral_-pi^pi (3 - 7 sin^2 x) cos