Solve the problem without using the Cauchy theorem.
Show that if \(\abs{a} < r < \abs{b}\), then \[ \int_\gamma \frac{dz}{(z-a)(z-b)}\,dz = \frac{2\pi i}{a-b}. \] Here \(\gamma\) denotes the circle centered at the origin, of radius \(r\), with the positive orientation.
This is part 3 of question 1 of the Spring 2014 list in the source, whose opening instruction is to solve the problem without using Cauchy’s theorem. The source prints \(dz\) twice in the integral.