## 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

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

Solution

1. Yes. It follows immediately from the identity theorem.
2. 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