Fractional-power integral from an upper-half-plane contour (${\int }_0^{\infty }{x}^s/(a^2+x^2)dx=\pi a^{s-1}/[2\cos (\pi s/2)],-1<s<1$)

Take $a>0$ and define $z^s$ using $-\pi /2<\arg z<3\pi /2$, so the branch cut lies on the negative imaginary axis. An indented upper semicircle encloses the simple pole $ia$. Its two straight segments contribute $(1+e^{i\pi s})$ times the desired real integral, while the residue theorem gives $\pi a^{s-1}{e}^{i\pi s/2}$. The large arc vanishes as its radius to the power $s-1$, and the small arc as its radius to the power $s+1$, giving the stated convergence range and result. This contour avoids placing a branch cut on either straight integration segment.