# What is the meaning of $dz$ in Complex Integrals?

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!

• Exactly the right question. – Tim kinsella Apr 16 '17 at 23:35
• You might be interested in this page. – DHMO Apr 16 '17 at 23:41

Suppose we parametrize the contour $C$ as $\gamma(t),\; a \le t \le b$. The contour integral $\int_C f(z)\; dz$ is the limit as $n \to \infty$ of "Riemann sums" $$\sum_{j=1}^n f(z_j)\; \Delta z_j$$ where $z_j = \gamma(t_j)$ and $\Delta z_j = z_j - z_{j-1}$, $a = t_0 < t_1 < \ldots < t_n = b$, if $\max(t_{j} - t_{j-1}) \to 0$.
• Again, I´m sorry for being too ignorant, but that "Δ" means difference right? In the sense that $Δzj$ is the difference between two "close" points in $C$ – Leo Apr 16 '17 at 23:58