How Do You Evaluate the Residues in the Cauchy Integral Formula for G_0(u)?

[marcusl]

I have a function
$$G_0=\frac{1-\alpha/u^2}{1-\alpha u^2} .$$

Since $$0<\alpha<1$$, $$G_0$$ has zeroes but no poles inside the unit circle.

I need to evaluate
$$\Gamma(\tau)=\frac{1}{2\pi i}\oint{\frac{\ln{G_0 (u)}}{u-\tau}du}$$
where the integral is around the unit circle. How do I evaluate the poles of the integrand so I can evaluate this using residues?

EDIT: Oops, G0 has a second order singularity at u=0, too.

 

[mathwonk]

maybe power series wouldwork, first the geometric series for G0 then the series for log(1+w)?