I'm sorry if I sound too ignorant, I don't have a high level of knowledge in math.
Interested in learning to integrate through other paths besides the real line in the complex plane, I searched how to do so in the internet. The most understandable "proof" of the "Line Integration Formula" was the following:
$C$ is a contour defined by $P(t)=R(t)+iI(t)$, $a<t<b$
$∫f(z)dz$ (through $C$) $=∫f(z(t))z'(t)dt$ (from $a$ to $b$)
since $z$ is in function of $t$, and $z'(t)=dz/dt$, so $dz$=$z'(t)dt$
The latter confuses me in the first step, when we state that the Integration of $C$ in $f$ $=∫f(z)dz$ (through $C$).
With the $∫f(z)$, I can imagine the sum of many (infinite) inputs of $C$ in $f$. But what is the meaning of $dz$?
In "real" integrals, $dx$ can be seen as $(b-a)/m$, where $m$ is the number of inputs we are adding, but I struggle when trying to see $dz$ in the same way since I can´t find the parallel of that $(b-a)$ in complex integrals. My first intuition was to think of $(b-a)$ as the length of the curve, but the "Line Integration Formula" seems to pay no attention on the length of the contour it is integrating.
Is there a way to approximate a complex integral in a similar way to the "real" integrals? I believe such approximation, if it exists, might give me an insight on the meaning of $dz$.
I think that my question is hard to explain, so if there is something that is confusing I would be happy to clarify.
I would really appreciate any explanation or insight!