Fourier Series: Odd and Even Functions Under the Interval (‒l, l)

ODD AND EVEN FUNCTIONS UNDER THE INTERVAL (‒l, l)

PROBLEMS UNDER THE INTERVAL (−l, l)

Example 1: If f(x) = x is defined in ‒l < x < l with period 2l, find the Fourier expansion of f(x).

Solution:

 f(x) = x

 f(‒x) = ‒x = ‒f(x)

Therefore f(x) is an odd function. Hence a₀ = 0 and a_n = 0

Let the required Fourier series be

Example 2: Obtain the Fourier series for the function given by

 f(x) is an even function. Hence b_n = 0

Let the required Fourier series be

Example 3: Expand f(x) = e^‒x as Fourier series in (‒1, 1).

Solution:

 f (‒x) = e^x ≠ f(x) and f(−x) = e^x ≠ −f (x)

Therefore f(x) is neither even nor odd

Let the required Fourier series be

Example 4: Find the Fourier series expansion of the periodic function

f(x) of period 2l defined by

f(x) = 1 + x, −1 ≤ x ≤ 0

= 1−x, 0 ≤ x ≤ 1

Deduce that 

Sol.

Given interval is (‒1, l)

Let f(x)

where ϕ₁(x) = 1+x, ϕ₂(x) = l‒x

Here ϕ₁(‒x) = 1 − x = ϕ₂(x)

f(x) is an even function.

Let the required Fourier series be

Example 5: Find the Fourier series expansion the following periodic function of period 4, f(x) = 

Hence, deduce that 

Solution: In the above problem

EXERCISE

Problems under the interval (−l, l)

1. Obtain the F.S. for f(x) defined in (‒1, 1) by

2. Find the F.S. for the function

3. f(x) is defined in (‒2, 2) as follows. Express f(x) in a F.S. of periodicity 4.

4. Find F.S. of periodicity 2 for f(x) given 

5. Find F.S. of periodicity 2 for

and hence deduce that 1 ‒ 1/3 + 1/5 ‒ 1/7 +….

6. Expand f(x) = x‒x² as a F.S. in −1 <x< 1.

7. Obtain the F.S. to represent x² from x = −l to x = l.

 [Ans. 4l² / n²π² (−1)ⁿ]

8. Find the F.S. of period 2 for the function

9. Find the F.S. of f(x) = x² +x in (‒2, 2). Hence find the sum of the series 1/1² + 1/2² + 1/3² +... + ∞

10. Find the Fourier expansion of f(x) = sin ax in (‒l, l)

11. Obtain the Fourier series of period 2l for the function f(x) = |x| in l≤x≤ l.

12. Find the Fourier Series of the following function

13. Find the Fourier series expansion of f(x) in (‒2, 2) which is defined as follows.