## On the dubious function

The *dubious function * is defined as follows : and

Evaluate the sum

## Definite logarithmic integral

Let . Evaluate the integral:

**Solution**

We have successively:

## Definite parametric integral

Let . Evaluate the integral

**Solution**

The key substitution is . Applying it we see that

Thus,

Thus , .

## Power of matrix

Let . Prove that .

**Solution**

The characteristic polynomial of is . This in return means and . Thus,

## Gamma infinite product

Prove that

**Solution**

Converting the product to a sum and using duplication formula for the gamma function and telescoping,

Using Stirling formula

we get that