## No rational function

Prove that there exists no rational function such that

Solution

Suppose , on the contrary , that such function exists. Since the harmonic series diverges we conclude that the limit of our function in infinity is infinity. This, in return means that the degree of the nominator , call that is greater that the one of the denominator , call that . Extracting in the nominator we get that

The limit of at infinity is finite and call that . Hence:

and of course this contradicts the fact that

where is the EulerÂ  – Mascheroni constant .

## A convergent series

Let be holomorphic on the open unit disk and suppose that the integral converges. If the Taylor expansion of is of the form then prove that the sum

converges.

Solution

We evaluate the integral using the standard orthogonality results for . Thus:

and thus the series converges.

Note: The set of functions satisfying this is a Hilbert space of functions, but it is not the same as the Hardy space .

## On Ahmed’s integral

Evaluate the integral

Solution

Let us begin by the identity

Thus

Now one of the ‘s definition is

Hence

Finally,

## Integral with trigonometric and exponential

Let . Prove that

Solution

Let us start things off by recalling the famous identity

which is a simple application of residues.

Thus making use of parity we have that

## Non existence of sequence of continuous functions

Prove that there does not exist a sequence of continuous functions such that converges pointwise, to the function , where is the characteristic polynomial of the rationals in .

Solution

The indicator of the rationals is no other function than

It is known that pointwise limits of continuous functions have a meagre set of points of continuity. However, this function is discontinuous everywhere and thus we cannot expect a sequence of continuous functions to converge pointwise to it.