Higher Engineering Mathematics

Assign-19-H8152.tex 23/6/2006 15: 18 Page 704

Fourier series

Assignment 19

This assignment covers the material contained

in Chapters 69 to 74.

The marks for each question are shown in

brackets at the end of each question.

1. Obtain a Fourier series for the periodic function
   f(x) defined as follows:

f(x)=

{

−1, when−π≤x≤ 0

1, when 0 ≤x≤π

The function is periodic outside of this range with

period 2π. (13)

2. Obtain a Fourier series to representf(t)=tin
   the range−πto+π. (13)


3. Expand the function f(θ)=θ in the range
   0 ≤θ≤πinto (a) a half range cosine series, and
   (b) a half range sine series. (18)


4. (a) Sketch the waveform defined by:

f(x)=

{

0, when− 4 ≤x≤− 2

3, when− 2 ≤x≤ 2

0, when 2 ≤x≤ 4

and is periodic outside of this range of period 8.

(b) State whether the waveform in (a) is odd,

even or neither odd nor even.

(c) Deduce the Fourier series for the function

defined in (a). (15)

5. Displacementyon a point on a pulley when
   turned through an angle ofθdegrees is given by:

θ y

30 3.99

60 4.01

90 3.60

120 2.84

150 1.84

180 0.88

210 0.27

240 0.13

270 0.45

300 1.25

330 2.37

360 3.41

Sketch the waveform and construct a Fourier

series for the first three harmonics (23)

6. A rectangular waveform is shown in Figure
   A19.1.
   (a) State whether the waveform is an odd or even
   function.
   (b) Obtain the Fourier series for the waveform in
   complex form.
   (c) Show that the complex Fourier series in (b) is
   equivalent to:

f(x)=

20

π

(

sinx+

1

3

sin 3x+

1

5

sin 5x

+

1

7

sin 7x+···

)

(18)

− 2 π

− 5

0 2 π

5

3 π x

f(x)

−π π

Figure A19.1