Fundamental Theorem of Calculus for Complex Riemann Integrals

Theorem
Let $\closedint a b$ be a closed real interval.

Let $F, f: \closedint a b \to \C$ be complex functions.

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

Then the complex Riemann integral of $f$ satisfies:


 * $\displaystyle \int_a^b \map f t \rd t = \map F b - \map F a$

Proof
Let $u, v: \closedint a b \times \set 0 \to \R$ be defined as in the Cauchy-Riemann Equations:


 * $\map u {t, y} = \map \Re {\map F z}$


 * $\map v {t, y} = \map \Im {\map F z}$

where:
 * $\map \Re {\map F z}$ denotes the real part of $\map F z$
 * $\map \Im {\map F z}$ denotes the imaginary part of $\map F z$.

Then: