Assignment:
By considering the integral of (z^2+1)^(-a) around a suitable contour C, prove:
Integral from x=0 to x=infinity of dx/(x^2+1)^a = sin(pi*a) Integral from u=1 to u=infinity of du/(u^2-1)^a
where 1/2 < a < 1.
(Include proofs that the integrals over any large or small circular arcs tend to zero as their radii tend to infinity or zero, whichever applies. Observe that (z^2 + 1)^(-a) has branch points at z = ±i and z = infinity.)
Provide complete and step by step solution for the question and show calculations and use formulas.