complexcharacter
I thought I'd try my hand at this and see what you all think. So here's the problem:
Let $z_0 \in \mathbb{C}$, and $r>0$. Then, let $u=z-z_0$, which implies $d u = dz$. This shifts the path from $C_r(z_0)$ to $C_r(0)$. Now, parametrize this path by $u=\gamma(t) = r e^{i t}$, with $t \in [0,2\pi)$. The integral
$$
\int_{C_r(z_0)} \frac{1}{z-z_0} \; dz
$$
is thus equal to
$$
\int_{0}^{2\pi} \frac{1}{\gamma(t)} \gamma'(t) \; dt = \int_{0}^{2\pi} \frac{1}{r e^{it}} \left(i r e^{it}\right) \; dt
$$
$$
= \int_{0}^{2\pi} i \; dt = 2\pi i.
$$
What do you guys think?
