My textbook in complex analysis defines a complex integral in the following way
Let $a, b \in \mathbb R$ and $g : [a, b] \to \mathbb C$ be a continuous complex-valued function. If $u$ and $v$ are both real-valued Riemann integratable functions over $[a, b]$ and $g(z) = u(z) + iv(z)$, then we define
$\displaystyle\int_a^b g(t) \; \mathrm dt := \displaystyle\int_a^b u(t) \; \mathrm dt + i \displaystyle\int_a^b v(t) \; \mathrm dt$
Let now make an alternative definition
Let $[a, b]$ be partitioned into $[a, b] = \displaystyle \bigcup_{k=1}^n \left[x_k, x_{k+1}\right]$ with $\begin{cases}x_1 = a \\ x_{k+1} > x_k \\ x_{n+1} = b\end{cases}$
Let $\Psi$ and $\Phi$ be complex valued functions such that
$\begin{cases}\mathcal{Re}(\Psi(z)) \leq \mathcal{Re}(f(z)) \leq \mathcal{Re}(\Phi(z)) \\ \mathcal{Im}(\Psi(z)) \leq \mathcal{Im}(f(z)) \leq \mathcal{Im}(\Phi(z)) \\ \Psi \text{ and } \Phi \text{ constant on } (x_k, x_{k+1}) \end{cases}$
Define the integral of $\Psi$ and $\Phi$ over $[a, b]$, denoted $I(\Psi)$ and $I(\Phi)$, as
$I(\Psi) := \displaystyle\sum_{k=1}^n \Psi(\xi_k) (x_{k+1}-x_k)$ where $\xi_k \in (x_k, x_{k+1})$
$I(\Phi) := \displaystyle\sum_{k=1}^n \Phi(\xi_k) (x_{k+1}-x_k)$ where $\xi_k \in (x_k, x_{k+1})$
We say that $g$ is Riemann integratable over $[a, b]$ if and only if $\forall \varepsilon > 0 \; \exists \Psi, \Phi : |I(\Phi) - I(\Psi)| < \varepsilon$
The unique complex constant $z_0 \in \mathbb C$ satisfying $\begin{cases}\mathcal{Re}(I(\Psi)) \leq \mathcal{Re}(z_0) \leq \mathcal{Re}(I(\Phi)) \\ \mathcal{Im}(I(\Psi)) \leq \mathcal{Im}(z_0) \leq \mathcal{Im}(I(\Phi)) \end{cases}$ for all functions $\Psi$ and $\Phi$ is called the integral of $g$ over $[a, b]$. We denote this number $\displaystyle\int_a^b g(t) \; \mathrm dt$.
Are these definitions equivalent? Can one extend the definition(s) above to compact sets $G \subseteq \mathbb C$ where $\partial G \subseteq G$ is a null set?