Questions And Answers

PART-A QUESTIONS AND ANSWERS

I. Problems under (0, 2), (0, 2), (-л, л), (−1, 1)

1. When does a function possess a Fourier series expansion in terms of trignometric terms?

(or)  State the conditions for f(x) to have Fourier series expansion.

(or) Explain Dirichlet's conditions.

Solution :

2. State whether y = tan x can be expanded as a Fourier series. If so how? If not why ?

Solution: tanx cannot be expanded as a Fourier series. Since, tanx does not satisfy Dirichlet's conditions.

[tanx has infinite number of infinite discontinuities.]

3. Find the sum of the Fourier series for

4.  If the Fourier series for the functiones.

5. Find the constant term in the Fourier series corresponding to

6. Write a_o, a_n in the expansion of x + x³ as a Fourier series in (-л, π).

7. What are the constant term ao and the coefficient of cos nx, a_n in the Fourier series expansion of

f (x) = x − x3 in (−ñ, ñ) ?

Solution: f(x) = x-x³

8. In the Fourier expansion of

9. If f(x) = x² + x is expressed as a Fourier series in the interval (-2, 2) to which value this series converges at x = 2?

Solution : x = 2 is a point of discontinuity in the extremum.

10. Find b1 in the expansion of as a Fourier Series in (-л, л).

Solution :

11. If f(x) is an odd function defined in (-1, 1), what are the values of a₀ and a_n

Solution: Given f(x) is an odd function in the interval (-1, 1)

12. Find the Fourier constants b_n for x sin x in (-л, л).

14. Determine the value of a_n in the Fourier series expansion of f(x) = x3 in −л < x <л.

15. If f (x) = 2x in the interval (0, 4), then find the value of a₂in the Fourier series expansion.

II. Problems under Half range series

1. Find half range sine series for f (x) = k in 0 < x <л.

Solution: The sine series of f (x) in (0, л) is given by

2.(a) Sketch the even and odd extension of the periodic function

f(x) = x2 for 0 < x <2

Solution :

2.(b) Sketch the graph of one even and one odd extension of ƒ (x) = x³ in [0, 1]

3. The cosine series for f (x) = x sin x for 0 < x <л is given as

4. Expand f(x) = 1 in a sine series in 0 < x < л.

Solution: The sine series of f (x) in (0, л) is given by

5. Find the Fourier sine series of f (x) = x in 0 < x < 2.

Solution: In the interval 0 < x < 2 the half range sine series for

6. To which value, the half range sine series corresponding to f(x) = x² expressed in the interval (0, 2) converges at x = 2?

Sol. Given f(x)2 

III. Problems under Parseval's identity, R.M.S value

1. State Parseval's identity for the half-range cosine expansion of f(x) in (0, 1).

Solution: 

2 Find the root mean square value of the function f(x) = x in the interval (0, 1).

3. Define root mean square value of a function f(x) in a < x < b.

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

4. What do you mean by Harmonic Analysis?

Solution: The process of finding Euler constant for a tabular function is known as Harmonic Analysis.   

5. Find the root mean square value of f (x) = x (1-x) in 0 ≤ x ≤l.

Solution :

6. Define R.M.S value of a function f (x) in c <x<c + 21.

Solution :

7. State TRUE or FALSE : Fourier series of period 20 for the nod function f(x) = x cos (x) in the interval (-10, 10) contains only sine terms. Justify your answer.

Solution :