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# Tag Archives: Special Functions

## A logarithmic integral

Evaluate the integral

Solution

Recall the idenity

thus,

Using Gautschi’s Inequality it follows that

and hence the integral equals

## On the factorial

Let denote the Möbius function and denote the floor function. Prove that:

Solution

The RHS equals

since for .

## Sum over all positive rationals

For a rational number that equals in lowest terms , let . Prove that:

Solution

First of all we note that

Moreover for we have that

Hence for we have that

## Bessel series

Let denote the Bessel function of the first kind. Prove that:

Solution

The Jacobi – Anger expansion tells us that

Hence by Parseval’s Theorem it follows that:

## On the dubious function

The dubious function is defined as follows : and

Evaluate the sum

### Who is Tolaso?

Find out more at his Encyclopedia Page.