Question

Problem 1 Expand the periodic functions (defined on the interval T. In a sine cosine ouer seres. (a) f(x) = (1+r) for-r < x < π f(x) for 0 -π<x < 0 for sinx 0<x<π = (c) f(x) ez for-r < x < π

Answer 1

(a) F(x) = 1 + x -pi < x < pi rightdoublearrow series for F(x) = 1 + x is given by f(x) = a_n + summationsign^infinity_n = 1 a_n cos n x + b_n sin n x where a_n = 1/pi integral^pi_-pi (1 + x)dx. = 1/pi [x + x^2/2]^pi_pi 1/pi[pi + pi^/2 -(-pi + pi^2/20] = 1/pi[2 pi] a_n = 2 a_n = 1/pi integral^pi_-pi (x + x) cosn x dx a_n = 1/pi { integral^pi_-pi cos n x dx + integral^pi_-pi x cos nx dx} = 1/pi{[sin n x/n]^pi_-pi + [x.sin n x/n]^pi_-pi - integral^pi_-pi sin n x/n dx} = {0 + 0 + [cos nx/n^2]^pi_-pi = 1/pi n^2 [(-1)^n - (-1)^n] therefore a_n = 0 \r\n bn = 1/pi integral^pi_-pi(1 + n).sin n x dx bn = 1/ pi{ integral^pi_-pi sin n x dx + integral^pi_-pi x sin n x dx} = 1/pi {-[cos n x]^pi_-pi/2