Prove that
Solution
Lemma 1: Let . It holds that
Lemma 2: Let . It holds that
We begin by squaring the identity of lemma 2. Hence,
Integrating the last equation we get,
Expanding the LHS we get that
Finally,
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Prove that
Solution
Lemma 1: Let . It holds that
Lemma 2: Let . It holds that
We begin by squaring the identity of lemma 2. Hence,
Integrating the last equation we get,
Expanding the LHS we get that
Finally,
Prove that
Solution
We have successively:
since for it holds that
Prove that
Solution
Since we have successively:
Let . Prove that
Solution
We are basing the whole solution on the Beta function and its derivative. We recall that
(1)
Setting and back at we get that
Differentiating with respect to we get that
where we made use of the reflection formulae of both the Gamma and the digamma function; and .
Now for our integral we have successively:
Prove that
Solution
We state two lemmata first.
Lemma 1: Let . It holds that
Proof: A standard proof can be found through Fourier Series. One can expand in Fourier series the function . Another way to prove the identity is to begin from the Weierstrass product , that is . Taking log on both sides we have that
Differentiating we have
Lemma 2: It holds that
Proof: Just differentiate the above identity.
Apply the above lemmata , we have for the initial integral that