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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: