Fourier Series: Odd and Even Functions

ODD AND EVEN FUNCTIONS

Certain functions defined in symmetric ranges of the form (‒π, π), (‒l, l) can be classified as even and odd functions.

Note

1: Functions defined in (−π, π) and (−l, l) may be either even or odd.

2. Functions defined in non‒symmetric range like (0, 2π), (0,2l) the case even or odd does not arise.

Definition:

 f(x) is said to be an even function of x in (−l,l) if f(−x) = f(x) Geometrically, the graph of an even function will be symmetrical with respect to Y axis.

Example: cosx, |x |, x², x sinx

Definition:

f(x) is said to be an odd function of x in (‒l,l) if f (−x) = −f (x) Geometrically, the graph of an odd function will be symmetrical about the origin.

Example: sinx, x³, x cos x

Example 1: What are the values of the Fourier constants when an even function f(x) is expanded in a Fourier Series in the interval ‒π to π ?

Solution:

The expansion will be of the form f(x) = a₀/2 + a_ncos nx

Example 2: What are the values of the Fourier constants when an odd function f(x) is expanded in a Fourier Series in the interval ‒π tо π ?

Solution:

The expansion will be of the form f(x) =  b_n sin nx

Example 3: What are the values of the Fourier constants when an even function f(x) is expanded in a Fourier Series in the interval ‒l to l ?

Solution:

The expansion will be of the form f(x) = 

Example 4: What are the values of the Fourier constants when an odd function f(x) is expanded in a Fourier series in the interval ‒l to l ?

Solution:

The expansion will be of the form 

Example 5: What are the values of the Fourier constants when f(x) is neither even nor odd in a Fourier series in the interval ‒π tо π?

Solution:

 Let the Fourier Series for f (x) in (−π, π) bе

Formulas (1), (2) and (3) are known as the Euler formulas.

Example 6: What are the values of the Fourier constants when f(x) is neither even nor odd in a Fourier series in the interval ‒l to l  ?

Solution:

The Fourier expansion for f(x) in the interval ‒1 < x < 1 is given by

Example 7: Find b_n in the expansion of x² as a Fourier Series in (− π, π).

Solution:

Given f(x) = x² is an even function in the interval (‒ π, π)

:. b_n = 0

Example 8: If f(x) is an odd function defined in (‒l, l), what are the values of a₀ and a_n ?

Solution:

Given f(x) is an odd function in the interval (‒l, l)

a₀=0, a_n= 0

Example 9: Find the Fourier constants b_n for x sin x in (‒π, π).

Solution:

Given f(x) = x sinx in (‒π, π)

 f(‒x) = (‒x) sin (‒x)

= (‒x) [‒ sin x]

= x sin x = f(x)

 f(x) is an even function

Hence b_n = 0

Example 10: Determine the value of a_n in the Fourier series expansion of f(x) = x³ in ‒π < x < ̧ π.

Solution:

Let f(x) = x³

f(‒x) = (‒x)³ = ‒x³ = f(x)

Therefore f(x) is an odd function.

Hence a₀ = 0 and a_n = 0

 Note: (1) If f(x) = 

such that (a) ϕ₁(−x) = −ϕ₂(x) or ϕ₂(−x) = −ϕ₁(x) then f(x) is said to be an even function of x in (‒π, π) (or) (−l,1)

(b) ϕ₁(−x) = −ϕ₂(x) or ϕ₂(−x) = −ϕ₁(x) then f(x) is said to be an odd function of x in (‒π, π) (or) (−l,1)

Example 11: Classifiy the functions as even, odd or neither.

Example 12: Classifiy the functions as even, odd or neither.