I am trying to teach myself the residue theorem, and I think I am close to putting all of the pieces together. However, I am struggling with the ML inequality as it pertains to arcs that appear in my contour.
For example, consider:
$\int_{-\infty}^{\infty} \frac{x^{1/2}}{(x^2 + 1)^2}dx$ and $\int_{-\infty}^{\infty} \frac{1}{(x^2 + 1)^2 x^{1/2}}dx$
I can properly draw the contour, find the residues, and deal with the parts of the contour along the real axis. However, I cannot find the integrals of the circular arcs around the origin, such as:
$\int\frac{1}{(z^2 + 1)^2 z^{1/2}}dz$
which should approach $0$ as we let the radius of the arc go to either $0$ or $\infty$. I can find "solutions" to the overall problems, but nothing that goes into enough detail for me to follow finding the "M" in the ML inequality.
Large radii: bound modulus of integrand from above by $2R^{-9/2}$.
Small radii: bound modulus of integrand from above by $2r^{-1/2}$