Let . Prove that:

**Solution**

We recall the following Fourier series.

**Lemma 1:** Let then

(1)

**Lemma 2:** Let then

(2)

**Lemma 3:** It holds that:

(3)

Hence for

since is whenever is even. Hence,

This is nothing else than Legendre function directly associated with the series in , . Hence,

The real part of is equation whereas the real part of is equation . Thus,

Because the LHS is even so must be the RHS. So, the result can be extended for . Therefore,

Another integral similar to the above. Let and . It holds that:

Proof:We have successively:Let us prove equation . Consider the function . Then:

On the other hand,

Equating the real parts we have that: