Linear Combination of Complex Integrals

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

Let $f, g: \closedint a b \to \C$ be complex Riemann integrable functions over $\closedint a b$.

Let $\lambda, \mu \in \C$ be complex constants.

Then:


 * $\displaystyle \int_a^b \paren {\lambda \map f t + \mu \map g t} \rd t = \lambda \int_a^b \map f t \rd t + \mu \int_a^b \map g t \rd t$

Proof
First, we prove the result for addition only without multiplying by $\lambda, \mu$:

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

By definition of complex number, we can find $\lambda_x, \lambda_y \in \R$ so $\lambda = \lambda_x + i \lambda_y$.

Then:

It follows from the results above that: