Calculate $\int_{0}^{\infty}\frac{x^a}{(x+b)^2}dx,|a|<1,b>0$
So I thought setting $f(z)=\frac{e^{aLog(z)}}{(z+b)^2}$ and using keyhole contour to calculate.
I managed to show that the integrals on the bigger and smaller circle are going to 0
but I got into trouble when i tried to set $z=re^{ai(2\pi-\epsilon)}$ and then plugging it into $f$
$f(re^{ai(2\pi-\epsilon)})=\frac{r^ae^{ai(2\pi-\epsilon)}}{(r^{i(2\pi-\epsilon)}+b)^2}=r^af(z)$(when $\epsilon\to 0$ )
but when I take $r\to\infty$ to calculate the integrals it doesn't make sense.
I'm supposed to calculate it with keyhold contour and then Residue Theorem
