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Background

Suppose $R(x)$ is a rational function with no positive real roots, and we want to integrate the following integral:

$$\int_0^\infty \frac{\log^n(x)}{R(x)} dx.$$

There's a trick that I've commonly seen where the integrand is multiplied by $\log(x)$ and this is integrated over $\gamma$, the keyhole contour together with the residue theorem.

$$\int_\gamma \frac{\log^{n+1}(z)}{R(z)} dz = 2\pi i\sum_{i}\operatorname{Res}_{z_i}\left(\frac{\log^{n+1}(z)}{R(z)}\right).$$


Question.

Is there a name for this technique?

I see on Wikipedia that there is a related example called "The square of the logarithm", but I'm looking for a term that I can look up in the index of a Complex Analysis text.

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    $\begingroup$ I've used the methodology you described many times to evaluate this class of integrals, but I've never seen a "name" attached to it. $\endgroup$
    – Mark Viola
    Jun 28, 2018 at 17:57

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