Understanding Engineering Mathematics

There are a number of points that are worth noting, although we do not want to be too

picky at this stage, so treat these as refinements for (much) later assimilation.

(i) At a discontinuity (e.g. att=0 in the above example) the Fourier series gives the

’average value’:

1

2 [f−(t^0 )+f+(t^0 )]

wheref−(t 0 )denotes the limiting value off(t)astapproaches the discontinuity at

t=t 0 frombelow,f+(t 0 )denotes the limit astapproaches fromabove.

(ii) The Fourier series of anevenfunction can contain only theconstantand thecosine

terms, because cosine is an even function. The Fourier series of anoddfunction can

contain only thesineterms, since sine is odd.

(iii) Interesting results for infinite series can be deduced from Fourier series. In the above
example, puttingt=π/2gives:

f

(π

2

)

=A=

4 A

π

(

1 −

1

3

+

1

5

−···

)

or 1−

1

3

+

1

5

−

1

7

+···=

π

4

(iv) While the Fourier series represents a periodic function, it can be used to represent

a non-periodic function just over one period if required. For example the square

wave considered above can be represented by a Fourier series over just one period

−π<t<π.

Exercise on 17.9

Sketch the triangular wave

f(t)=t 0 <t<π

=−t −π<t< 0

f(t)=(t+ 2 π)

for− 5 π<t< 5 π.

Is the function odd or even? The corresponding Fourier series is

f(t)=

π

2

−

4

π

(

cost+

1

32

cos 3t+···+

1

( 2 r+ 1 )^2

cos( 2 r+ 1 )t+···

)

(you will be asked to derive this in Section 17.10). What is the average value of the function

over all time (consider the average value of any sinusoid over a complete period)? What

is the fundamental component? Write down the amplitude of thenth harmonic.

Deduce from the series that

1 +

1

32

+

1

52

+···+

1

( 2 r+ 1 )^2

+···=

π^2

8