\begin{definition}[Definition:Complex Riemann Integral]
Let $\mathbb I := \closedint a b$ be a closed real interval.
Let $f: \mathbb I \to \C$ be a bounded complex function.
Define the real function $x: \mathbb I \to \R$ by:
:$\forall t \in \mathbb I: \map x t = \map \Re {\map f t}$
Define the real function $y: \mathbb I \to \R$ by:
:$\forall t \in \mathbb I: \map y t = \map \Im {\map f t}$
Here:
:$\map \Re {\map f t}$ denotes the real part of the complex number $\map f t$
:$\map \Im {\map f t}$ denotes the imaginary part of $\map f t$.
Suppose that both $x$ and $y$ are Riemann integrable over $\mathbb I$.
Then the '''complex (Riemann) integral of $f$ over $\mathbb I$''' is defined as:
:$\ds \int_a^b \map f t \rd t = \int_a^b \map \Re {\map f t} \rd t + i \int_a^b \map \Im {\map f t} \rd t$
$f$ is formally defined as '''(properly) complex integrable over $\mathbb I$ in the sense of Riemann''', or '''(properly) complex Riemann integrable over $\mathbb I$'''.
More usually (and informally), we say:
:'''$f$ is (Riemann) complex integrable over $\mathbb I$.'''
\end{definition}