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# Tag Archives: Special Functions

## 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

## Divergent Möbius series

Let denote the Möbius function. Prove that

Solution

Summing only over primes , where ,  we have that

## Application of extended binomial theorem

For the values of for which the following series makes sense, prove that

Solution

We have successively:

due to the extended binomial theorem.

## Double summation

Evaluate the sum

Solution

Lemma: It holds that

Proof: Consider the function and let us it integrate over the following contour

By the residue theorem it follows that

For the residues we have

The integrals along the sides vanish; hence:

and since

the result follows.

Back to the problem. We have successively: