Complex Riemann Integral is Contour Integral

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Let $f: \R \to \C$ be a complex Riemann integrable function over some closed real interval $\closedint a b$.


$\ds \int_a^b \map f t \rd t = \int_\CC \map f t \rd t$


the integral on the left hand side is a complex Riemann integral
the integral on the right hand side is a contour integral
$\CC$ is a straight line segment along the real axis, connecting $a$ to $b$.


\(\ds \int_a^b \map f t \rd t\) \(=\) \(\ds \int_a^b \map f {\map \theta t} \map {\theta'} t \rd t\) Complex Integration by Substitution: $\map \theta t = t$, $\map {\theta'} t = 1$
\(\ds \) \(=\) \(\ds \int_\CC \map f t \rd t\) Definition of Complex Contour Integral