Explanation of path integrals

[redshift]

What exactly does a path integral measure? Is it area between the ends/bounds of the line? Or is it the length of the line? Just started complex analysis and am comletely confused by this.

[shmoe]

What exactly does a path integral measure? Is it area between the ends/bounds of the line? Or is it the length of the line? Just started complex analysis and am comletely confused by this.

It's not the length, you can think of the dz as keeping track of direction and distance you move. The integral \int_{\gamma}dz along a path z=\gamma(t),\ t\in\[a,b\] is simply \int_{a}^{b}\gamma '(t)dt=\gamma(b)-\gamma(a) so you actually get the distance and direction your end point is from your start point, not the total distance you've travelled. This integral doesn't even keep track of how the path made it from start to finish, nor how fast it went.