Fourier Series Of Even And Odd Functions
Definition : A function f is said to be an even function if f(x) = f(-x) for all x and it is odd if f(-x) = -f(x) . Examples : Sin kx , x,x³ and any power of x are all odd functions where as Cos kx , x ,1,x² and any even power of x are even functions . The following properties of even and odd functions are easy to check a a (i) ∫ f(x) = 0 if f is odd = 2∫ f(x) dx -a 0 if f is even . (ii) The product of even and odd function are characterised by even * odd = odd even * even = even odd * odd = even . Now suppose f is an even periodic function with period 2π . Then since the sine function is odd , f(x) sin nx is odd and f(x) cos nx is even . So by (i) π bₙ = 1/π ∫ f(x) sin nx = 0 ,