Integral ∫1−11x√1+x1−xlog((r−1)x2+sx+1(r−1)x2−sx+1)dx\int_{-1}^{1} \frac{1}{x}\sqrt{\frac{1+x}{1-x}} \log \left( \frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} – sx + 1} \right) \, \mathrm dx

Regarding this problem, I conjectured that


Though we may try the same technique as in the previous problem, now I’m curious if this generality leads us to a different (and possible a more elegant) proof.

Indeed, I observed that I(r,0)=0 and


which can be evaluated using standard contour integration technique. But simplifying the residue and integrating them seems still daunting.

EDIT. By applying a series of change of variables, I noticed that the problem is equivalent to prove that


for π2<α<π2 and s>1. (This is equivalent to the condition that the expression inside the logarithm is positive for all xR.)

Another simple observation. once you prove that ˜I(α,s) does not depend on the variable s for s>1, then by suitable limiting process it follows that


which (I guess) can be calculated by hand. The following graph may also help us understand the behavior of this integral.

enter image description here


So, following the procedure I outlined here, I get for the transformed integral:


Note that this reduces to the integral in the original problem when r=3 and s=2. Then we see that the roots of the denominator satisfy the same symmetries as before, so we need only find one root of the form ρeiθ where




Using the same methodology I derived, I am able to confirm your conjecture.

Source : Link , Question Author : Sangchul Lee , Answer Author : Community

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