When evaluating real integrals involving log, I am having trouble with the step that involves finding a bound on circular segments. Let me explain what I mean:
If, for example, we have $ \int_0^\infty \frac{(\log(x))^2}{1+x^2} \, \mathrm{d}x $ We consider the complex integral $ \int\frac{(\log(z))^2}{1+z^2} \, \mathrm{d}z $ along a path on which the function is analytic. In this case, our path, gamma, would be made of four segments:
- from $\epsilon$ to $R$ along the positive real axis,
- from $R$ to $-R$ along a circle in the upper half plane
- from $-R$ to $-\epsilon$ on the negative real axis
- from $-\epsilon$ to $\epsilon$ along a circle in the upper half plane
(in this way we can consider the branch of log excluding the negative imaginary axis)
I understand that you then proceed to show that integrals 2 and 4 reduce to zero as $R$ approaches infinity and $\epsilon$ approaches zero. This is where I have trouble. Most resources simply say, "show f is bounded".
What is the typical procedure for finding a bound for this type of function involving log? (Or even not involving log.)
I'm sorry for the messy latex and I would be very appreciative of any guidance you can provide.