Higher Engineering Mathematics, Sixth Edition

636 Higher Engineering Mathematics

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

f(t)=

{

t, 0 <t< 1

( 2 −t), 1 <t< 2

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

f(t)=

8

π^2

(

sin

(

πt

2

)

−

1

32

sin

(

3 πt

2

)

+

1

52

sin

(

5 πt

2

)

−···

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

4. Show that the half-range Fourier cosine series
   for the functionf(θ )=θ^2 in the range 0 to 4
   is given by:

f(θ )=

16

3

−

64

π^2

(

cos

(

πθ

4

)

−

1

22

cos

(

2 πθ

4

)

+

1

32

cos

(

3 πθ

4

)

−···

)

Sketch the function within and outside of the

given range.