Contour Integral of Concatenation of Contours

Theorem
Let $C$ and $D$ be contours.

Suppose that the endpoint of $C$ is equal to the start point of $D$, so the concatenation $C \cup D$ is defined.

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

Then:


 * $\displaystyle \int_{C \cup D} f \left({z}\right) \ \mathrm dz = \int_C f \left({z}\right) \ \mathrm dz + \int_D f \left({z}\right) \ \mathrm dz$

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 $i \in \left\{ {1, \ldots, n}\right\}$.

Similarly, $D$ is a finite sequence $D_1, \ldots, D_m$ of directed smooth curves.

Let $D_j$ be parameterized by the smooth path $\sigma_j: \left[{c_j\,.\,.\,d_j}\right] \to \C$ for all $j \in \left\{ {1, \ldots, m}\right\}$.

Put $\gamma_i = \sigma_{i-n}, a_i = c_{i-n}$ and $b_i = d_{i-n}$ for all $i \in \left\{ {n+1, \ldots, n+m}\right\}$.

Then $C \cup D$ is a sequence of $n+m$ directed smooth curves which are parameterized by $\gamma_1, \ldots, \gamma_{n+m}$.

Then: