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# Tag Archives: Special Functions

## Multiple logarithmic integral

Let denote the Riemann zeta function. Evaluate the integral

Solution

Based on symmetries,

Let . It follows that

Using the recursion we get that

Thus,

## An integral equality

Let denote the trigamma function and let be integrable on . It holds that

Solution

We have successively:

where the interchange between the infinite sum and the integration is allowed by the uniform bound

The result follows.

## An series related to the eta Dedekind function

Prove that

Solution

Let us consider the function

and integrate it along a quadratic counterclockwise contour with vertices where is a big odd natural number. Hence,

We note that and that .

It’s also easy to see that

Hence, as we have that

By Residue theorem we have that

It is straightforward to show that

Hence,

in the limit . The result follows.

## Bessel function integral

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

Solution

We recall that

Hence,

Then,

Using the fact that the looks like an ‘almost periodic’ function with decreasing amplitude. If we denote by the zeros of then as and furthermore

as for each . So the integral converges uniformly in this case justifying the interchange of limit and integral.

The result follows.

## On multiplicative functions

Let denote the Möbius function and the Euler’s totient function. Prove that

Solution

Since both functions are multiplicative it suffices to prove the identity for prime numbers. Hence for we have