Higher Engineering Mathematics

EVEN AND ODD FUNCTIONS AND HALF-RANGE FOURIER SERIES 675

L

or f(x)=

8

π

(

1

3

sin 2x+

2

( 3 )( 5 )

sin 4x

+

3

( 5 )( 7 )

sin 6x+···

)

Now try the following exercise.

Exercise 243 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

sin 2x

−

1

9

sin 3x

−

π

8

sin 4x+···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

cos 3t

+

1

5

cos 5t−···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

(b) f(t)=

2

π

(

sint−sin 2t

+

1

3

sin 3t+

1

5

sin 5t

−

1

3

sin 6t+···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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)

−

sin 3x

(1)(3)(5)

−

sin 5x

(3)(5)(7)

−

sin 7x

(5)(7)(9)

−···

)

(b) f(x)=

1

2

(1−cos 2x)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

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

f(x)=

⎧

⎪⎪

⎨

⎪⎪

⎩

x,0<x<

π

2

(π−x),

π

2

<x<π

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

f(x)=

π

4

−

2

π

(

cos 2x

+

cos 6x

32

+

cos 10x

52

+···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦