Complex Riemann Integral is Contour Integral

Theorem
Let $f: \R \to \C$ be a complex Riemann integrable function over some closed real interval $\left[{a\,.\,.\,b}\right]$.

Then:
 * $\displaystyle \int_a^b f \left({t}\right) \, \mathrm d t = \int_\mathcal C f \left({t}\right) \, \mathrm d t$

where:
 * the integral on the is a complex Riemann integral
 * the integral on the is a contour integral
 * $\mathcal C$ is a straight line segment along the real axis, connecting $a$ to $b$.