What is the complex line integral measuring? How do we motivate it? How do we motivate it?
I am not necessarily looking for a geometric answer to my question so much as I am looking for a way to motivate the idea of a complex line integral. For any path $\gamma$ and any function $f(z)$ we have
$$\int_\gamma f(z)=\int f(\gamma(t))\gamma '(t)dt$$
and it turns out that this has a lot of power in terms of characterizing functions.
But historically how did we make a decision to define things this way? What motivates this? I've gotten a lot of after the fact answers but I don't really feel like this is satisfactory. It turns out this is useful doesn't motivate its original discovery/definition very well.
Possible questions:
What is the complex line integral measuring?
 A: Here's how these things were motivated (in a very loose but still pretty historically accurate sense):
People started playing around with an idea: the integrals people were using at the time were tied to intervals.  Intervals are just paths in complex space, and so when you start trying to understand what integrals mean in complex space, it's natural to look at what it would mean to do integrals in complex paths.
Great, that's kind of interesting, and you immediately get the equation you post when you start looking at parameterizing the path.  So there's a fairly easy to investigate it too.  If nothing else happened, that would all be fairly interesting, but also pretty trivial.  Complex analysis could move on with integrals in complex space and the relation wouldn't be given much thought.
But something else does happen.  Loops detect singularities and extract differential structure.  It was found that you could actually use these integrals to extract a lot of important information from analytic functions that describe the nature of the functions.  So, once people started to play with path integral, the motivations exploded.
Good Complex Analysis books or courses will go over this in quite a lot of detail.
