Odd And Even Functions

ODD AND EVEN FUNCTIONS

Certain functions defined in symmetric ranges of the form (-π, π), (-1, 1) can be classified as even and odd functions. 0)

Note 1: Functions defined in (-л, л) and (-1, 1) may be either even or odd.

2. Functions defined in non-symmetric range like (0, 2л), Jedi (0,27) the case even or odd does not arise.

Definition:

f(x) is said to be an even function of x in (-1, 1) 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 sin x

Definition :

f(x) is said to be an odd function of x in (-1,1) 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 2.2.1 What are the values of the Fourier constan when an even function f(x) is expanded in a Fourier Series in the interval -πιο π?

Solution :

Example 2.2.2: What are the values of the Fourier constants when an odd function f(x) is expanded in a Fourier Series in the interval –π tο π?

Solution :

Example 2.2.3: What are the values of the Fourier constants when an even function f(x) is expanded in a Fourier Series in the interval -l tol ?

Solution :

Example 2.2.4: What are the values of the Fourier constants when an odd function f (x) is expanded in a Fourier series in the interval -1 to 1 ?

Solution:

Example 2.2.5: What are the values of the Fourier constants when f(x) is neither even nor odd in a Fourier series in the interval -л to π?

Example 2.2.6: 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 :

Example 2.2.7: Find b in the expansion of as a Fourier Series in (-л, л).

Solution :

Example 2.2.8: If f(x) is an odd function defined in (-1, 1), what are the values a₀ of a_n ?

Solution :;

Example 2.2.9 Find the Fourier constants b for x sin x in (-л, л).

Example 2.2.10: Determine the value of an in the Fourier series expansion of f (x) = ^x3 in - < x < л.

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

Example 2.2.11 Classifiy the functions as even, odd or neither.

Example 2.2.12. Classify the following functions as even, odd or  neither.