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# Tag Archives: Special Functions

## 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: ## Arithmotheoretic sum

Evaluate the sum Solution

The sum converges absolutely , so we can switch the order of summation; hence: The last sum equals and hence ## Gamma inequality

Let be three positive real numbers such that . Prove that: where is Euler’s Gamma function.

Solution

We can rewrite the inequality as since is convex.

## Asymptotic expansion

Prove that for some constant .

## A product

Let denote the golden ratio , the Möbius function and Euler’s totient function. Prove that: Solution

We are using the following facts:

(1) (2) Hence, ### Donate to Tolaso Network 