Definition:Complex Riemann Integral

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Definition

Let $\left[{a \,.\,.\, b}\right]$ be a closed real interval.

Let $f: \left[{a \,.\,.\, b}\right] \to \C$ be a bounded complex function.


Define the real function $x : \left[{ a \,.\,.\, b }\right] \to \R$ by:

$\forall t \in \left[{ a \,.\,.\, b }\right] : x \left({ t }\right) = \operatorname{Re} \left({ f \left({ t }\right) }\right)$

Define the real function $y : \left[{ a \,.\,.\, b }\right] \to \R$ by:

$\forall t \in \left[{ a \,.\,.\, b }\right] : y \left({ t }\right) = \operatorname{Im} \left({ f \left({ t }\right) }\right)$

Here:

$\operatorname{Re} \left({ f \left({ t }\right) }\right)$ denotes the real part of the complex number $f \left({ t }\right)$
$\operatorname{Im} \left({ f \left({ t }\right) }\right)$ denotes the imaginary part of $f \left({ t }\right)$.


Suppose that both $x$ and $y$ are Riemann integrable over $\left[{a \,.\,.\, b}\right]$.


Then the complex (Riemann) integral of $f$ over $\left[{a \,.\,.\, b}\right]$ is defined as:

$\displaystyle \int_a^b f \left({t}\right) \ \mathrm d t = \int_a^b \operatorname{Re} \left({ f \left({t}\right) }\right) \ \mathrm d t + i \int_a^b \operatorname{Im} \left({ f \left({t}\right) }\right) \ \mathrm d t $


$f$ is formally defined as (properly) complex integrable over $\left[{a \,.\,.\, b}\right]$ in the sense of Riemann, or (properly) complex Riemann integrable over $\left[{a \,.\,.\, b}\right]$.


More usually (and informally), we say:

$f$ is (Riemann) complex integrable over $\left[{a \,.\,.\, b}\right]$.


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