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Poisson like Integral

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,

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

Proof: We have successively:

2. Let us prove equation . Consider the function . Then:

On the other hand,

Equating the real parts we have that: