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I am trying to figure out if the following integrals are equivalent by some sort of symmetry $$f_1(z)=\int_{\max\{-1-z,-1\}}^{\min\{1-z,1\}}\frac{dy}{\pi^2 \sqrt{1-(z+y)^2}{\sqrt{1-y^2}}}$$ and the following

$$f_2(z)= \int_{\max\{z-1,-1\}}^{\min\{z+1,1\}} \frac{dy}{\pi^2 \sqrt{1-(z-y)^2}{\sqrt{1-y^2}}}$$

Thanks for any help.

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Observe that the difference between the two integrands is the change of variable $y\mapsto-y$. Now look at the boundaries of your integrals to check if they match after the precedent change of variables (use the fact that $-\min\left(a,b\right)=\max\left(-a,-b\right)$ and conversely).

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  • $\begingroup$ thanks but that doesnt help in my case right? $\endgroup$
    – Henry
    Feb 18, 2015 at 23:39
  • $\begingroup$ Indeed, it seems that there should be a factor $-1$. $\endgroup$
    – Nicolas
    Feb 18, 2015 at 23:49

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