## Multiple integral

Let denote the usual inner product of . Evaluate the integral

where is a positive symmetric matrix and .

Solution

Since is a positive symmetric matrix , so is . For a positive symmetric matrix there exists an positive symmetric matrix such that . Applying this to our integral becomes

where is the Euclidean norm. Applying a change of variables we have that

Since then by converting to polar coordinates we have that

Here denotes the surface area measure of the unit sphere and it is known to be

hence

where denotes the Gamma Euler function for which it holds that

## Limit of a trigonometric integral

Let be a measurable of finite length set. Evaluate the limit:

Solution

Let us begin with the Fourier series of which is of the form:

Hence

(1)

Integrating ( 1 ) we get that

since it follows from Riemann – Lebesgue lemma that

## A generating function involving harmonic number of even index

Let denote the -th harmonic number. Prove that forall it holds that

Solution

Well we are stating two lemmata.

Lemma 1: For all it holds that

Proof: Pretty straight forward calculations show that

and Lemma 1 is proved.

Lemma 2: For all it holds that

Proof: We begin by lemma 1 and successively we have

and Lemma 2 follows.

Now, mapping back at Lemma we have that

Integrating we have that

## Constant function

Let be a continuous function such that

where . Prove that is constant.

Solution

Well it does hold

where and . The set is dense in and since is continous and characterised by its values in a dense set we conclude that is constant since

## On a known pattern

Let and consider the function

where and . If for all then prove that

Solution

The function is differentiable in and the derivative is given by

Hence we want to prove that . We note that forall it holds that

Thus

and the conclusion follows.