# Is there some difference between the two definition of integration along curves about complex function

In Stein's Complex Analysis, the integral of $f$ along $\gamma$ is defined by $$\int_{\gamma}f(z)\text{d}z=\int_a^bf(z(t))z'(t)\text{d}t$$

where $z:[a,b]\rightarrow \Bbb{C}$ is a parametrization of $\gamma$

the right hand of the equation is a definite integral, and it is defined by: $$\int_a^bf(z(t))z'(t)\text{d}t =\lim_{||\Delta t||\rightarrow 0} \sum_{k=1}^{n}f(z(\xi_k))z'(\xi_k)(t_{k}-t_{k-1})$$

where $a=t_0<t_1<\cdots<t_{n-1}<t_n=b$ is a segmentation of $[a,b]$ and $t_{k-1}<\xi_k<t_k$ and $||\Delta t||=\max\{|t_{1}-t_{0}|,|t_{2}-t_{1}|,\cdots,|t_{n}-t_{n-1}|\}$

If I want to define the integral by
$$\lim_{||\Delta z||\rightarrow 0} \sum_{k=1}^{n}f(\zeta_k)(z_k-z_{k-1})$$ where $z_k$ is points on $\gamma$ which divided $\gamma$ into some segments

assume $f$ is continuous and $z'(t)$ is continuous too. then they are all exists and equal.

My Question is: why author define $\int_{\gamma}f(z)\text{d}z=\int_a^bf(z(t))z'(t)\text{d}t$,but not directly use Riemann sum to define:$\int_{\gamma}f(z)\text{d}z=\lim_{||\Delta z||\rightarrow 0} \sum_{k=1}^{n}f(\zeta_k)(z_k-z_{k-1})$?

thanks very much

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Because the second form is not quite a Riemann sum. You have terms like $z_k - z_{k-1}$ there, which are now complex numbers, i.e. two-dimensional real vectors. It's impossible to take subdivision of these (how do you subdivide a vector?), so you'd need to build whole Riemann integration theory for complex numbers from scratch. This seems like a waste of time for most applications.
On the other hand, the first form reduces to the usual real Riemann integration, so you can directly apply those results in this situation. The only price paid in doing so is that the definition now depends on the parametrization $z(t)$ and one needs to prove that the integral is actually reparametrization invariant.