Linear Combination of Complex Integrals

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

Let $f, g: \left[{a \,.\,.\, b}\right] \to \C$ be complex Riemann integrable over $\left[{a \,.\,.\, b}\right]$.

Let $w, z \in \C$ be complex numbers.

Then:


 * $\displaystyle \int_a^b \left({w f \left({t}\right) + z g \left({t}\right) }\right) \ \mathrm dt = w \int_a^b f \left({t}\right)  \ \mathrm dt + z \int_a^b g \left({t}\right)  \ \mathrm dt$

Proof
First, we prove the result for addition only without multiplying with $w, z$:

Next, we show the result for only one complex integral multiplied by a constant $z$.

By definition of complex number, we can find $x, y \in \R$ so $z = x + iy$.

Then:

It follows from the results above that: