## 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

### Who is Tolaso?

Find out more at his Encyclopedia Page.