Solution 8

(1)

(a)
It is periodic with period 2. So, consider segment between .

Let

be the Fourier expansion, where:

For ,

But, in

Substituting in x(t)

---

(b)
It is periodic with period 6. So, consider segment between .

The function in this interval is:

 

 

 

Let

be the Fourier expansion, where:

For ,

Substituting in x(t)

 

---

(c)
It is periodic with period 3, so take the segment

The function in the interval is:

Let

be the Fourier expansion, where:

For ,

 

Substituting in x(t)

 

---

(d)
  It is periodic with period 2. So take segment

Here,

Let

be the Fourier expansion, where:

For

Substituting in x(t)

 

---

(e)
It is periodic with period 6. So, take the segment

Here will be

Let

be the Fourier expansion, where:

For

Substituting in x(t)

 

---

(f)
It is periodic with period 3. So, take the segment

Here will be

Let

be the Fourier expansion, where:

For

Substituting in x(t)

(2) Let

be the Fourier expansion, where:

Given .

For

Substituting in x(t)

(3) Let

be the Fourier expansion, where:

Given .

For

Substituting in x(t)