## An integral with trigonometric and rational function

Evaluate the integral

**Solution**

Let us now consider the function as well as the contour

The function has three simple poles of which only is included within the contour. The residue at turns out to be . Thus

The contribution of the large circle as is whereas the contribution of the small circle as is . This can be seen by parametrising the small circle ( ). Hence:

Using we get that

## A logarithmic integral

Prove that

**Solution**

Now let us consider the complex function where the principal arguement of lies within the interval as well as the contour below

It is clear that has two poles of order at and . The residue at is equal to whereas the residue at is equal to . Thus

Sending and the contribution of both the large and the small circle is . Hence:

Thus the conclusion follows.

## Is f necessarily constant?

Let be a differentiable function such that forall . Does it follow that is necessarily constant?

**Solution**

contradicting what we had assumed in the first place. Hence is constant.

## A contour integral

Let such that . Prove that:

**Solution**

## Convergence of series

Examine if the series:

converges.

**Solution**

as well as the Hermite Hadamard inequality. Applying the Hermite – Hadamard inequality we have that

Exponentiating we get that

and hence

This leads us to the conclusion that the series diverges.