Fourier Series: Dirichlet's Conditions Under the Interval (0, 2π)

FOURIER SERIES DIRICHLET'S CONDITIONS UNDER THE INTERVAL (0, 2π)

IMPORTANT FORMULAE

Note:

1. Bernoulli's formula: ʃuv dx = uv₁ ‒ u'v₂ + u"v₃ ‒, …. Where u and v are functions of x.

4. (‒1)ⁿ⁺¹ = (‒1)^n‒1

5. cos nπ = (−1)ⁿ, cos 2nπ = (−1)²ⁿ = [(−1)²]ⁿ = [1]ⁿ = 1

sin nπ = 0, sin 2nπ = 0 if n is an integer.

PROBLEMS UNDER THE INTERVAL (0, 2π)

FOURIER SERIES DIRICHLET'S CONDITIONS UNDER THE INTERVAL (0, 2π) – EXERCISE PROBLEMS

Exercise

Problems under the interval (0, 2π)

1. Show that in the range 0 to 2π the Fourier series expansion for

2. Express f(x) = 1/12 (3x²‒6xπ+2π²) as a Fourier series of period 2π in the interval (0, 2π). Hence, show that

3. An alternating current after passing through a rectifier has the form 

where i₀ is the maximum current and the period is 2π.

Express 'i' in a Fourier series.

4. Find the Fourier series of

5. Expand f (x) = αx (π‒x) as a Fourier series of period 2π in the 0 ≤x≤2π when a is a constant.

6. Express f(x) = (x‒x)² as a F.S of period 2π in the interval 0 < x < 2π. Hence, deduce the sum of the series

7. Express f(x) = 

8. Prove that 1/12 x (π‒x) (2π‒x) = Σ_odd^∞[ Sin nx / n³ ]

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

f(x) = 1/12 x (π − x) (2π − x) in (0, 2π),

Deduce the sum of the series 1/1³ ‒ 1/3³ + 1/5³ ‒ ….

 [Ans. 1/12 x (π − x) (2π − x) = ∑₁^∞ 1/n³ sin nx; π³/32]

10. Find the Fourier series of f(x) = 1/4 (π − x)² in 0 < x < 2π.

Hence, deduce that 1/1² + 1/2² +1/3² + …. = π²/6

 [Ans. f(x) =  ]

11. Find the Fourier series to represent f(x)=

12. Find the Fourier series to represent x‒x² in the interval (0, 2π)

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

f(x) = x cos x in 0 < x < 2π.

 [Ans. f(x)= ]

14. Derive the Fourier expansion of √(1‒cosx) in 0 ≤x≤2π and deduce that 

.

15. Find the value of a₀ in the Fourier series expansion of f (x) = e^x in (0, 2π). [Ans. a₀ = 1/π [e^2π ‒ 1]