Fundamental Theorem of Calculus for Complex Riemann Integrals

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

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

Suppose that $F$ is a primitive of $f$.

Then:


 * $\displaystyle \int_a^b f \left({t}\right) \ \mathrm dt = F \left({b}\right) - F \left({a}\right)$

Proof
Let $u, v: \left[{a \,.\,.\, b}\right] \times \left\{ {0}\right\} \to \R$ be defined as in the Cauchy-Riemann Equations:


 * $u \left({t, y}\right) = \operatorname{Re} \left({F \left({z}\right) }\right)$


 * $v \left({t, y}\right) = \operatorname{Im} \left({F \left({z}\right) }\right)$

Here, $\operatorname{Re} \left({F \left({z}\right)}\right) $ denotes the real part of $F \left({z}\right)$, and $\operatorname{Im} \left({F \left({z}\right)}\right) $ denotes the imaginary part of $F \left({z}\right)$.

Then: