Fourier Series: Parseval's Relation (or) Theorem (or) Identity

PARSEVAL'S RELATION (or) THEOREM (or) IDENTITY

Let f(x) be a periodic function with period 2π defined in the interval (‒π, π).

Then 

 where a₀, a_n and b_n are Fourier coefficients of f(x).

Proof:

The Fourier series of f(x) defined in (‒π, π) is given by

Multiplying (1) by f(x) and integrating term by term from ‒π to π.

Note: If f(x) is a periodic function in c <x<c+2l with period

Definition: Root Mean Square Value [RMS Value] (or) Effective value

Let f(x) be a function defined in an interval (a, b) then

is called the root mean square (or) effective value of f(x) and is denoted by .

Hence 

Note: Parseval's theorem gives the values of root mean square of f (x) in terms of its Fourier coefficients.

Problems based on Parseval's theorem

Example 1: Obtain the Fourier series expansion of f(x) = x² in (‒l, l). Find the sum of [ 1/1⁴ + 1/2⁴ + 1/3⁴ + …. ].

Solution:

f(‒x) = f(x)

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

Example 2: Find the sine series for f(x) = x in 0 < x < π. Using R.M.S. value, show that π²/6 = 1 + 1/2² + 1/3² + …..

Solution: See the value b_n in Example 1.

Example 3: Find the Fourier series x² in (‒π, π). Use Parsevals identity to prove π4/90 = 1 + 1/24 + 1/34 + 1/44 + …...

Solution:

the values of a₀, a_n and b_n in Example 2.

By Parseval's theorem

Example 4: Find the cosine series for f(x) = x in (0, π) and then using Parseval's theorem.

Show that 

Solution:

the values of a₀ and a_n in Example 6.

Example 5: Find the half range cosine series of f(x) = (x − x²) in the interval (0,π). Hence find the sum of the series 1/1⁴ + 1/2⁴ + 1/3⁴ + ... + ∞.

Solution: the values in Example 7

Example 6: Find the half‒range cosine series for the function f(x) = x (π − x) in 0 < x < π. Deduce that 

Solution: See the values in Example 8

EXERCISES

1. Expand f(x) = x ‒ x² as a Fourier series in ‒1 < x < 1 and using this series find the r.m.s value of f (x) in the interval.

2. Prove that in 0<x<l; x= 

Hence deduce that 

3. Find the Fourier series of period 2π for the function

and hence find the sum of the series

4. Obtain the sine series for f(x) = c in the interval 0 < x < π.

Hence prove that 1/1² + 1/3² + ... = π²/8