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# Tag Archives: Integral

## Exponential fractional part

Let denote the fractional part of . Evaluate the integral Solution

By definition it holds that hence We note however that Hence, Summing up, ## Arctan squared integral

Evaluate the integral Solution

First of all we note that for  Hence by Parseval we get that Consider the branch of which is defined outside the segment and which coincides with the positive square root for . Let then evaluate the contour integral: Solution

It is a classic case of residue at infinity. Subbing the counterclockwise contour integral rotates the northern pole of the Riemannian sphere to the southern one and the contour integral is transformed to a clockwise one. Hence: The equality does hold for all if we take the standard branch , otherwise it is not that obvious why this holds, since we are dealing with a multi-valued function.

## Square logarithmic integral

Prove that Solution

Lemma 1: Let . It holds that Lemma 2: Let . It holds that We begin by squaring the identity of lemma 2. Hence, Integrating the last equation we get, Expanding the LHS we get that Finally, ## A parametric logarithmic integral

Let . Prove that Solution

We are basing the whole solution on the Beta function and its derivative. We recall that

(1) Setting and back at we get that Differentiating with respect to we get that where we made use of the reflection formulae of both the Gamma and the digamma function; and .

Now for our integral we have successively: ### Donate to Tolaso Network 