User:Anghel/Sandbox

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

Then:


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

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

Let $z \in \C$.

Then:


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

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

Then: