Linear Combination of Contour Integrals

Theorem
Let $C$ be a contour.

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

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

Then:


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

Proof
By definition of contour, $C$ is a finite sequence $C_1, \ldots, C_n$ of directed smooth curves.

Let $C_i$ be parameterized by the smooth path $\gamma_i: \left[{a_i\,.\,.\,b_i}\right] \to \C$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

Then: