Triangle Inequality for Contour Integrals

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Theorem

Let $C$ be a contour.

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


Then:

$\displaystyle \left\vert{ \int_C f \left({z}\right) \ \mathrm dz }\right\vert \le \max_{z \mathop \in \operatorname{Im} \left({C}\right) } \left\vert{f \left({z}\right) }\right\vert L \left({C}\right)$

where $L \left({C}\right)$ denotes the length of $C$.


Proof

By definition of contour, $C$ is a concatenation of 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:

\(\displaystyle \left\vert{ \int_C f \left({z}\right) \rd z }\right\vert\) \(=\) \(\displaystyle \left\vert{ \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t}\right\vert\) Definition of Complex Contour Integral
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \left\vert{ \int_{a_i}^{b_i} f \left({\gamma_i \left({t}\right) }\right) \gamma_i' \left({t}\right) \rd t}\right\vert\) Triangle Inequality for Complex Numbers
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \left\vert{ f \left({\gamma_i \left({t}\right) }\right) }\right\vert \left\vert{ \gamma_i' \left({t}\right) }\right\vert \rd t\) Modulus of Complex Integral
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \max_{t \mathop \in \left[{a_i \,.\,.\, b_i}\right]} \left\vert{f \left({\gamma_i \left({t}\right) }\right) }\right\vert \int_{a_i}^{b_i} \left\vert{ \gamma_i' \left({t}\right) }\right\vert \rd t\) Linear Combination of Integrals
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \max_{z \mathop \in \operatorname{Im} \left({C}\right)} \left\vert{f \left({z}\right) }\right\vert \int_{a_i}^{b_i} \left\vert{ \gamma_i' \left({t}\right) }\right\vert \rd t\) as $\gamma_i \left({t}\right) \in \operatorname{Im} \left({C}\right)$
\(\displaystyle \) \(=\) \(\displaystyle \max_{z \mathop \in \operatorname{Im} \left({C}\right) } \left\vert{f \left({z}\right) }\right\vert L \left({C}\right)\) Definition of Length of Contour (Complex Plane)

$\blacksquare$


Sources