Contour Integral is Independent of Parameterization

Theorem
Let $C$ be a contour defined by 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 $i \in \left\{ {1, \ldots, n}\right\}$.

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

Suppose that $\sigma_i: \left[{c_i\,.\,.\,d_i}\right] \to \C$ is a reparameterization of $C_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

Then:


 * $\displaystyle \int_C f \left({z}\right) \ \mathrm dz = \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right)  \ \mathrm dt = \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} f \left({\sigma_i \left({t}\right) }\right) \sigma_i' \left({t}\right)  \ \mathrm dt$

Proof
By definition of parameterization, $\gamma_i = \sigma_i \circ \phi_i$ for all $i \in \left\{ {1, \ldots, n}\right\}$.

Here, $\phi_i: \left[{c_i \,.\,.\, d_i}\right] \to \left[{a_i \,.\,.\, b_i}\right]$ is a bijective differentiable strictly increasing function.

Then: