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# Tag Archives: Integral

## 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, ## Fourier transformation

Let and . Show that Solution

We note that Thus, ## Arctan integral

Prove that where denotes the Euler – Mascheroni constant.

Solution

Beginning by parts we have, However, The result now follows taking and .

## 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.

## Trigonometric integral

Evaluate the integral Solution

We have successively: ### Donate to Tolaso Network 