How can I show the following, for $n\geq 0$:
$$ \frac{1}{2\pi} \oint_{\ \Gamma} \frac{1}{z} \left(z + \frac{1}{z}\right)^{2n} dz $$
using a contour $\Gamma$ defined as the unit circle centered at the origin and oriented counterclocwkise.
Ref. Complex Analysis by M.W. Wong
The residue theorem asks for the $z^{-1}$ term of $z^{-1}(z+z^{-1})^{2n}$, which is the same as the constant coefficient of $(z+z^{-1})^{\color{Red}{2n}}$. If I said we get $\binom{\color{Red}{2n}}{n}$ would you know how the binomial appears?