= Fractional-power integral from an upper-half-plane contour
{title2=$\int_0^\infty x^s/(a^2+x^2)\,dx=\pi a^{s-1}/[2\cos(\pi s/2)],\quad -1<s<1$}
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.
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