Contour Integral is Independent of Parameterization

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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:

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


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:

\(\displaystyle \int_C f \left({z}\right) \rd z\) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t\) Definition of Complex Contour Integral
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \int_{\phi_i^{-1} \left({a_i}\right) }^{\phi_i^{-1} \left({b_i}\right)} f \left({\gamma_i \left({\phi_i \left({u}\right) }\right) }\right) \gamma_i' \left({\phi_i \left({u}\right) }\right) \phi_i' \left({u}\right) \rd u\) substitution with $t = \phi \left({u}\right)$
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \int_{\phi_i^{-1} \left({a_i}\right) }^{\phi_i^{-1} \left({b_i}\right) } f \left({ \sigma_i \left({u}\right) }\right) \sigma_i' \left({u}\right) \rd u\) Derivative of Complex Composite Function
\(\displaystyle \) \(=\) \(\displaystyle \sum_{i \mathop = 1}^n \int_{c_i}^{d_i} f \left({ \sigma_i \left({u}\right) }\right) \sigma_i' \left({u}\right) \rd u\) Reparameterization of Directed Smooth Curve Maps Endpoints To Endpoints

$\blacksquare$


Sources