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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: ### Donate to Tolaso Network 