SRI VENKATESWARA COLLEGE OF ENGINEERING DEPARTMENT OF APPLIED MATHEMATICS

181301 TRANSFORMS AND PARTIAL DIFFERENTIAL EQUATIONS

WORKSHEET / UNIT: I FOURIER SERIES

PART- A

1. State the conditions for a given function f(x) to satisfy for the existence of a

Fourier series.

2. Does tanx = f(x) possess a Fourier expansion?

3. If xxxf 2)( is expressed as a Fourier series in the interval )2,2( , to

which value this series converges at x = 2?

4. If f(x) is discontinuous at x = a, what does its Fourier series represent at that

point?

5. Give the expression for the Fourier series co-efficient bn for the function f(x)

defined in(-2,2).

6. If f(x) is an odd function defined in ),,( ll what are the values of ?0 naanda

7. Determine the value of na in the Fourier series expansion of 3)( xxf in

. x

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

expansion.

9. Define the RMS value of a function f(x) in )2,0( l .

10. The Fourier series expansion of f(x) in )2,0( is

1

sin)(

n n

nxxf . Find the

root mean square value of f(x) in the interval )2,0(

11. Find the root mean square value of 2)( xxf in the interval ),0( .

12. State Parseval’s identity for full range expansion of f(x) as Fourier series

in )2,0( l .

13. If the Fourier series corresponding to f(x) = x in the interval )2,0( is

1

0 },sincos{2

nxbnxaa

nn without finding the values of nn baa ,,0 find

the value of

1

22

2

0 ).(2

nn baa

[Ans.:3

8 2]

14. What do you mean by Harmonic Analysis?

[Ans.: 3) 4, 6) 00 naa , 7) 0, 8) 0, 10) 2

1

2

1

nf 11)

5

2f ]

PART – B

1. Expand the following as a Fourier series

a. x xf (x) sin in 20 x and deduce

that4

2...

7.5

1

5.3

1

3.1

1

.

b. |cosx |)( xf in x .

c. x)( 2xf in x . Hence prove . 3

1

2

11

90 44

4

d. x -in cos1(x) xf .

e. )(xf 21in )2(

10in x

xx

x

f.

lxlin

lxinxlxf

20

0)( . Hence deduce

that 4

....7

1

5

1

3

11

and .

8 .

5

1

3

1

1

1

2

222

g. .0,

0,)(

xx

xxxf Hence deduce that

1

2

2.

8)12(

1

n

h. .10,1

01,0)(

x

xxf i..

xexf )( in ).,( ll

[Ans.: (a)

222

1

1

cos2sincos1)(

n n

xxxxf

.

(b)

1 )12)(12(

2cos)!(42)(

n

n

nn

nxxf

. (c)

22

2 cos)1(2

3)(

n

n

n

nxxf

.

(d)

1 )12)(12(

cos2422)(

n nn

nxxf

. (e)

5,3,1

2

cos4

2)(

n n

xnxf

.

(f)

1

22cos])1(1[

4)(

l

xn

n

llxf n

1

sinl

xn

n

l

.

(g) .cos])1(1[2

2)(

12

n

n nxn

xf

.sin}])1(1{1

[2

1)()

1

n

n xnn

xfh

i)

1222

1222

sinsinh)1(2

cossinh)1(2sinh

)(n

n

n

n

l

xn

nnl

ln

l

xn

nnl

ll

l

lxf

]

2. Expand the following as half range Fourier Sine Series

a. )(xf

lxl

xl

lxx

2in

20in

[Ans.:l

xnn

n

l

n

sin

2sin

14

122

]

b. ).,0(cos)( inxxxf [Ans:

22

]sin1

2)1(sin

2

1)(

n

n nxn

nxxf

c. lxxxf 0in )( . Hence evaluate

1

2

2.

6

1

n

d. ).,()( loinaxf Deduce the sum of . . 5

1

3

1

1

1

222

[Ans.: c) .sin)1(12

)( 1

1 l

xn

n

lxf n

n

d)

oddn l

xn

n

axf ]sin

14)(

3. Expand the following as half range Fourier Cosine Series

a. 20in )2()( 2 xxxf . Deduce that

1

2

2.

6

1

n

b. xxxxf 0 ,sin)( . Hence show

that .4

2...

7.5

1

5.3

1

3.1

1

c. )()( xxxf in .0 x Deduce that .90

...3

1

2

1

1

1 4

444

[Ans.: (a)

1

22 2cos

16

3

4)(

xn

nxf

.

(b)

222

1

1

cos)1(2cos1)(

n

nxxxf

n

.

(c)

evenn

nxn

xf cos1

46

)(2

2]

4. Find the complex form of Fourier series for the function xexf )( in

.11 x

5. Find the complex form of Fourier series for the function xxf sin)( in

.0 x

6. If a is not and integer, find the complex Fourier series of axxf cos)( in

).,( [Ans.: (4)xin

n

n

en

inxf

221

1sinh)1()1()( .

(5)

n

nxien

xf 2

2 14

12)(

. (6) ]

)(2

sin2)1()(

22

inxn

ena

aaxf

7. The table value of the function )(xfy is given below. Find a Fourier series

up to the second harmonic to represent f(x) in terms of x.

a)

b)

[Ans.: (a) xxxf 2cos1.0cos367.045.1)( .2sin574.0sin173.0 xx

(b)3

2sin0266.0

3sin13.1

3

2cos33.2

3cos33.883.20)(

xxxxxf

]

8. For the following data show that sin004.1cos37.075.0)( xf where

.2

T

x

x 0 π /3 2 π/ 3 π 4 π/3 5 π/ 3 2 π

y 1.0 1.4 1.9 1.7 1.5 1.2 1.0

x 0 1 2 3 4 5

y 9 18 24 28 26 20

x 0 T/6 T/3 T/2 2T/3 5T/6 T

y 1.98 1.3 1.05 1.3 -0.88 -0.25 1.98