Higher Engineering Mathematics

662 FOURIER SERIES

and (b) the sum of the Fourier series at the

points of discontinuity.

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(a)f(x)=

1

2

+

2

π

(

cosx−

1

3

cos 3x

+

1

5

cos 5x−···

)

(b)

1

2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

f(x)

− 3 π

2

−π −π

2

π

2

π^3 π

2

(^0) x
1
Figure 69.6

4. For Problem 3, draw graphs of the first three
   partial sums of the Fourier series and show
   that as the series is added together term by
   term the result approximates more and more
   closely to the function it represents.


5. Find the term representing the third har-
   monic for the periodic function of period 2π
   given by:

f(x)=

{

0, when−π<x< 0

1, when 0 <x<π

[

2

3 π

sin 3x

]

6. Determine the Fourier series for the periodic
   function of period 2πdefined by:

f(t)=

⎧

⎪⎪

⎪⎨

⎪⎪

⎪⎩

0, when−π<t< 0

1, when 0 <t<

π

2

−1, when

π

2

<t<π

The function has a period of 2π

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

f(t)=

2

π

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

cost−

1

3

cos 3t

+

1

5

cos 5t−···

+sin 2t+

1

3

sin 6t

+

1

5

sin 10t+···

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

7. Show that the Fourier series for the periodic
   function of period 2πdefined by

f(θ)=

{

0, when−π<θ< 0

sinθ, when 0 <θ<π

is given by:

f(θ)=

2

π

(

1

2

−

cos 2θ

(3)

−

cos 4θ

(3)(5)

−

cos 6θ

(5)(7)

−···

)