Even and Odd functions

Even and Odd functions

A function f(x) is said to be even if f (-x) = f (x). For example x2, cosx, x sinx, secx are even functions. A function f (x) is said to be odd if f (-x) = - f (x). For example, x3, sin x, x cos x,. are odd functions.

(1) The Euler‟s formula for even function is a

(2) The   Euler‟s   formula   for   odd   function   is

Example 6

Find the Fourier Series for f (x) = x in ( -p, p)

Here, f(x) = x        is an odd function.

Example 7

Expand f (x) = |x| in (-p, p) as FS and hence deduce that

Example 8

Then find the FS for f(x) and hence show that

Here f (-x) in (-p,0) = f (x) in (0,p)

f  (-x) in (0,p) = f (x) in (-p,0) 

f(x) is a even function

Example 9

Obtain the FS expansion of f(x) = x sinx in (-p< x<p) and hence deduce that

Here f (x) = xsinx  is an even function.

Exercises:

Determine Fourier expressions of the following functions in the given interval:

i. f(x) = p/2 + x, -p< x < 0

p/2 - x, 0 < x < p

ii. f(x) = -x+1 for + -p< x < 0

x+1 for 0 =< x =< p