Questions and Answers

PART-A

QUESTIONS AND ANSWERS

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

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 :

See Page No. 2.13 

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

Sol : See Page No. 2.20

4. If the Fourier series for the function

Sol. Put x = π/2 is a point of continuity.

5. Find the constant term in the Fourier series corresponding to f(x) = cos2x expressed in the interval (— π, π).

Sol.

6. Write a0, an in the expansion of x+x3 as a Fourier series in (-л, π).

Sol. Let f(x) = x + x3

7. What are the constant term a0 and the coefficient of cos nx, an in the Fourier series expansion of f(x) = x − x3 in (−π, π) ?

Solution : f(x) = x - x3

8. In the Fourier expansion of

of bn, the coefficient of sin nx.

Solution :

⸫ Given function is an even function.

Hence, the value of bn = 0

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 bn in the expansion of x2 as a Fourier Series in (-π, π).

Solution :

Given f (x) = x2 is an even function in the interval (-л, л)

⸫ n = 0

11. If f(x) is an odd function defined in (−l, l), what are the values of a0 and an?

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

⸫ a0 = 0, an = 0

12. Find the Fourier constants b1 for x sin x in (-π, π).

Solution: 

13. If f(x) = and f(x) = f (x + 2 л) for all x, find the sum of the Fourier series of ƒ (x) at x = π.

Solution: 

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

Solution : 

15. If f (x) = 2x in the interval (0, 4), then find the value of a2 in the Fourier series expansion.

Solution: 

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 ƒ (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 f(x) = x3 in [0, 1]

Solution :

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

Solution :

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) = x2

x = 2 is a finite point of discontinuity and also it is an end point.

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, l).

Solution: 

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 ƒ (x) = x (l − x) in 0 ≤ x ≤ l.

Solution :

6. Define R.M.S value of a function ƒ (x) in c < x < c + 2l.

Solution :

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

Solution :

TRUE. f (x) = x cosx is odd in (-10, 10) = 0

⸫ a0 = 0 and an = 0

So, we get only sine terms.