Compute $\int_{\gamma} z\, dz$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$

Question

Compute the complex line integral$$\int\limits_{\gamma} z\, \mathrm{d}z$$ for any smooth path $\gamma$ which begins at $z_0$ and ends at $w_0$.

Confused as to how I am supposed to go about parametrizing any smooth path $\gamma$. I know how to solve this for a line from $z_0$ to $w_0$ but I am stumped by "any" smooth path.

Answer 1

$f(z)=z$ is an entire function in the complex plane, so $\int_\gamma z\,dz$ only depends on the endpoints of $\gamma$ and not the "interior" of it (like a conservative vector field). The result for any path from $z_0$ to $w_0$ is the same as the result for the straight line from $z_0$ to $w_0$: $\frac12(w_0^2-z_0^2)$.