Linear Combination of Contour Integrals

Theorem
Let $C$ be a contour in $\C$.

Let $f, g: \Img C \to \C$ be continuous complex functions, where $\Img C$ denotes the image of $C$.

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

Then:


 * $\ds \int_C \paren {\lambda \map f z + \mu \map g z} \rd z = \lambda \int_C \map f z \rd z + \mu \int_C \map g z \rd z$

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

Let $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.

Then: