## 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**