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Chapter: Mathematics (maths) - Fourier Series

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Important Questions and Answers: Fourier Series

Mathematics (maths) - Fourier Series - Important Short Objective Question and Answers: Fourier Series


 

FOURIER SERIES


 

1. Explain periodic function with examples.

 

A function  f (x)is said to have a period T if for all  xf (x +T )=f (x), where T is a

 

positive constant. The least value of T >0 is called the period of  f (x).

 

Example :  f (x)=sin x ;  f (x  +2p) sin=(x  2 +p) sin=x .


2. State Dirichlet’s conditions for a function to be expanded as a Fourier series.


Let a function f ( x) be defined in the interval c <x <c 2+p with period 2p and satisfies the following conditions can be expanded as a Fourier series in (c, c +2p) .

 

(i)             f (x) is a well defined function.

 

(ii)           f ( x) is finite or bounded.

 

(iii)        f ( x) has only a finite number of discontinuous point.

 

(iv)        f ( x) has only a finite number of maxima and minima.

 

 

 

 

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

 

tan x  cannot  be  expanded  as  a  Fourier  series.  Since  tan x  not   satisfies   D

 

condition.

 

 

4. State the convergence condition on Fourier series.

 

 

(i)    The Fourier series of f ( x) converges to f ( x) at all points where f ( x) is continuous.

 

 

 

(ii) At a point of discontinuity x0 , the series converges to the average of the left limit and right limit

 

of  f (x) at x0


 

5. To what value does the sum of Fourier series of  f (x) converge at the point of continuity

 

x   =a ?

 

The sum of Fourier series of  f (x) converges to the value  f (a) at the continuous point

 

x =a .

6. To what value does the sum of Fourier series of f ( x) converge at the point of discontinuity x =a ?

 

At the discontinuous point x =a , the sum of Fourier series of  f ( x) converges to




 


\  sinh x is an odd function.

 

 

\  a0  =0, an    =0.

 

 

 

 

10. Write the formulae for Fourier constants for  f ( x) in the interval (-p,  p).

 

The Fourier constants for  f (x) in the interval (-pp)are given by


11. Find the constant a0 of the Fourier series for function  f (x)=x in 0 £x £2p.


The given function f ( x ) = |x| is an even function.



14. Find bn in the expansion of x2 as a Fourier series in (-p,p).

 

Since  f ( x ) =x2  is an even function, the value of bn  =0

15. Find the constant term a0  in the Fourier series corresponding to  f (x )= x -x3   in

 

(-π, π).

 

Given f (x)=x -x3

 

f (-x)=-x +x3  =-(x-x3 )=-f (x)

 

i.e,      f (-x)=-f (x)

 

\                   f ( x) is an odd function in (-p,p)

 

Hence a0  =0 .

 

16. If  f (x)=x 2   -x4  is expanded as a Fourier series in (-l,l ), find the value of bn .

 


The coefficient of sin nx , bn =0 . Since the Fourier series of  f ( x)  consists of cosine terms only.

 

18. Find the constant a0 of the Fourier series for the function f (x) =  xcosx in - π < x < π.


-π < x <. Π

f ( x ) =x cos x

f ( -x )  =x-cos x    =f(x-)

 

\    f ( x) is an odd function.  Hence a0  =0 .

 

 

 

 

19.            Write the Fourier sine series of k in (0,p).

 


 

20. Obtain the sine series for unity in (0, π).




 

22. If  f (x)is defined in -3  £x  3£what is the value of Fourier coefficients.


 

23. Define Root Mean Square value of a function.

 

The root mean square value of  y =f ( x) in (a , b) is denoted by y . It is defined as


 

 

24. Find the R.M.S value of y =x 2   in (-p, p).


 

25. Find the R.M.S value if  f ( x ) = x2  in -π £x £.π


 

26. State   the   Parseval’s   Identity   (or)   theorem

 

If  f ( x)  is a periodic function of period 2p in  (c, c +2p) with Fourier coefficients


 

 

27. Write the complex form of Fourier series for f(x) defined in the interval (c, c+2l).

 

 

The  series  for   f (x)  defined  in  the  interval  (c, c +2p)  and satisfying Dirichlet’s  conditions can be given in the form of


 

28. What do you mean by Harmonic analysis?

 

 

The process of finding the Fourier series of the periodic function  y =f (x)of period 2l (or )2p using the numerical values of x and  yBar is known as Harmonic analysis.

 

 

 

PART B


 






 

2) Obtain Fourier series for f(x) of period 2l and defined as follows






 

Hence proved

 

 

 

 

 

ODD AND EVEN FUNCTION

 

 

3.  Find the Fourier series of  


Therefore f(x) is neither even nor odd function

 

We know that the Fourier series is




 

FOURIER SINE SERIES





This is the required half range Fourier sine series.

 

HALF RANGE COSINE SERIES

4)                Obtain the half range cosine series for  f(x)=(x-2)2 in the interval (0,2).

 

olution:

Given

 f(x)=(x-2)2


We know that the Fourier half range cosine series is








 This is the required Fourier series

 

COMPLEX FORM OF FOURIER SERIES

6) Find the complex form of the Fourier series of 


 



 

 

 

HARMONIC ANALYSIS

 

7)Computeupto first harmonics of the Fourier series of f(x) given by the following table

 

X       0        T/6    T/3    T/2    2T/3  5T/6  T

F(x)   1.98   1.3     1.05   1.3     -0.88 -0.25 1.98

Solution:                                                             

First and last value are same. Hence we omit the last value.            

When x varies from 0 to T                                            

          varies from 0 to 2p                                    

We know that the Fourier series is   






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