# Definition:Complex Riemann Integral

## Definition

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:

$\displaystyle \int_a^b \map f t \rd t = \int_a^b \map \Re {\map f t} \rd t + i \int_a^b \map \Im {\map f t} \rd t$

$f$ 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 $\mathbb I$.