Higher Engineering Mathematics

682 FOURIER SERIES

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣

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.