## Double summation

Evaluate the sum

**Solution**

**Lemma:** It holds that

*Proof: *Consider the function and let us it integrate over the following contour

By the residue theorem it follows that

For the residues we have

The integrals along the sides vanish; hence:

and since

the result follows.

Back to the problem. We have successively:

## On functions

- Does there exist an entire such that for all ?
- Does there exist a holomorphic function on the unit disk such that for all ?

**Solution**

- Yes. It follows immediately from the identity theorem.
- Νο, there is no such function. If there was, its Taylor series centered at would be of the form
for some and . But then

In particular

But

## Explicit formula of f

In the following figure the function is continuous and . For every point on the curve the areas of and are equal.

You are asked to find an explicit formula for .

**Solution**

We are expanding the above figure so that it looks like this;

The area is given by

On the other hand we have that

Differentiating with as a variable and dependent to we get that . Hence,

## Limit of a Riemann sum

Let be . Prove that

## Arithmotheoretic sum

Evaluate the sum

**Solution**

The sum converges absolutely , so we can switch the order of summation; hence:

The last sum equals and hence