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## Ahmed’s Integral

Prove that Solution

Consider the function . Differentiating with respect to we have that: We integrate the last equation from to . Thus, However, Hence the last equation gives

(1) Suffice to calculate the integral. Applying the change of variables we have: Going back at we have that: ## Coxeter’s Integral

Prove that Solution

We state lemmata:

Lemma 1: It holds that .

Lemma 2: It holds that .

Proof: We have successively: Lemma 3: It holds that where .

Proof: We have successively: We are ready to attack the initial monster. For that we have: ## On the centralizer

Suppose that has this property that if is an eigenvalue of then is not an eigenvalue of . Show that if and only if for any . In other words the centralizer of equals the centralizer of .

Solution

It is clear that implies for any . Now suppose that for some and set . We want to prove that . We have and so . It now follows that for any integer and thus for any and any integer we have where is the identity matrix. Now let be a generalized eigenvector corresponding to an eigenvalue of . Then for some integer and thus, by we have . Therefore, since we are assuming that is not an eigenvalue of , we must have . So, since every element of is a linear combination of some generalized eigenvectors of , we get for all , i.e. and hence .

The exercise can also be found here.

## Square of a number

Let such that . Prove that is rational.

Solution

Setting , and we note that . Hence, The result follows.

## A logarithmic integral

Evaluate the integral Solution

Recall the idenity thus, Using Gautschi’s Inequality it follows that and hence the integral equals ### Who is Tolaso?

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