Higher Engineering Mathematics, Sixth Edition

Even and odd functions and half-range Fourier series 629

Hence the half-range Fourier sine series forf(x)in the
range 0 toπis given by:

f(x)=

8

3 π

sin2x+

16

15 π

sin4x

+

24

35 π

sin6x+···

or f(x)=

8

π

(

1

3

sin2x+

2

( 3 )( 5 )

sin4x

+

3

( 5 )( 7 )

sin6x+···

)

Now try the following exercise

Exercise 231 Further problems on

half-range Fourier series

1. Determine the half-range sine series for the
   function defined by:

f(x)=

⎧

⎨

⎩

x, 0 <x<

π

2

0 ,

π

2

<x<π

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

f(x)=

2

π

(

sinx+

π

4

sin2x

−

1

9

sin3x

−

π

8

sin4x+···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

2. Obtain (a) the half-range cosine series and
   (b) the half-range sine series for the function

f(t)=

⎧

⎪⎨

⎪⎩

0 , 0 <t<

π

2

1 ,

π

2

<t<π

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(a) f(t)=

1

2

−

2

π

(

cost

−

1

3

cos3t

+

1

5

cos5t−···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡

⎢

⎢

⎢

⎢

⎢⎢

⎢

⎣

(b) f(t)=

2

π

(

sint−sin2t

+

1

3

sin3t+

1

5

sin5t

−

1

3

sin6t+···

)

⎤

⎥

⎥

⎥

⎥

⎥⎥

⎥

⎦

3. Find (a) the half-range Fourier sine series and
   (b) the half-range Fourier cosine series for the
   functionf(x)=sin^2 xin the range 0≤x≤π.
   Sketch the function within and outside of the
   given range.
   ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣
   (a) f(x)=

8

π

(

sinx

( 1 )( 3 )

−

sin3x

( 1 )( 3 )( 5 )

−

sin5x

( 3 )( 5 )( 7 )

−

sin7x

( 5 )( 7 )( 9 )

−···

)

(b) f(x)=

1

2

( 1 −cos2x)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

4. Determine the half-range Fourier cosine series
   in the rangex=0tox=πfor the function
   defined by:

f(x)=

⎧

⎪⎪

⎨

⎪⎪

⎩

x, 0 <x<

π

2

(π−x),

π

2

<x<π

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

f(x)=

π

4

−

2

π

(

cos2x

+

cos6x

32

+

cos10x

52

+···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦