Definition:Integrable Function/Complex
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Definition
Let $\mathbb I := \closedint a b$ be a closed real interval.
Let $f: \mathbb I \to \C$ be a bounded complex-valued 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:
- $\forall t \in \mathbb I: \map y t = \map \Im {\map f t}$
where:
- $\map \Re {\map f t}$ 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 $\mathbb I$.
Sources
- 2001: Christian Berg: Kompleks funktionsteori $\S 2.2$