Linear Combination of Contour Integrals

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

Let $\gamma: \left[{ a \,.\,.\, b }\right] \to \C$ be a contour.

That is, there exists a subdivision $a_0, a_1, \ldots, a_n$ of $\left[{ a \,.\,.\, b }\right]$ such that $\gamma \restriction_{I_i}$ is a smooth path for all $i \in \left\{ {1, \ldots, n}\right\}$, where $I_i = \left[{a_{i-1} \,.\,.\, a_i}\right]$.

Let $f, g: \operatorname{Im} \left({\gamma}\right) \to \C$ be continuous complex functions, where $\operatorname{Im} \left({\gamma}\right)$ denotes the image of $\gamma$.

Let $z_0 ,z_1 \in \C$ be complex numbers.

Then:


 * $\displaystyle \int_{\gamma} \left({ z_0 f \left({z}\right) + z_1 g \left({z}\right)} \right) \ \mathrm dz = z_0 \int_{\gamma} f \left({z}\right)   \ \mathrm dz + z_1 \int_{\gamma} g \left({z}\right)   \ \mathrm dz$