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This question has no detail answers and I would like to ask the similar question. I deal with the following integral: $$\int_{0}^{1}dx\,x(1-x) \ln\frac{M^2-k^2x(1-x)}{\lambda^2},$$ where one can assume that $M>0$, $\lambda>0$, ... (in other words, I would loke to emphasize that integral is convergent). The "branch point" is $$x_{1,2}=\frac{1}{2}\pm\frac{1}{2}\sqrt{k^2-4M^2}$$ and I don't know how to evaluate this integral. I think about integration in complex plane with suitable contour but don't know how to perform this calculation.

I know the answer with help of Mathematica but I am interested in how to do it "by hands". Also, may be this question this question is helpful for further discussion.

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    $\begingroup$ The integrand has an antiderivative in the form $P(x) + Q(x) \log \frac{M^2 - k^2 x(1-x)}{\lambda^2} + \tan^{-1} (ax + b)$ for some polynomials $P, Q$ and constants $a, b$. It's grungy but straightforward to verify. $\endgroup$
    – anomaly
    Jan 13 '19 at 15:33
  • $\begingroup$ @anomaly in Russian language I don't see word "antiderivative". If I consider indefinite integral, the answer for this indefinite integral is the "antiderivative"? What about degrees of $P$ and $Q$? $\endgroup$ Jan 13 '19 at 17:12
  • $\begingroup$ Yes. (Well, some people are pedantic about the implicit additive constant.) The degrees of $P$ and $Q$ are $2$ or $3$; I don't remember exactly what they are offhand. $\endgroup$
    – anomaly
    Jan 13 '19 at 17:15
  • $\begingroup$ Ok, now I know how to find anti-derivative but the answer is really compicated :D $\endgroup$ Jan 13 '19 at 18:20
  • $\begingroup$ That's QFT in a nutshell. :) $\endgroup$
    – anomaly
    Jan 13 '19 at 18:30

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