Triangle Inequality for Contour Integrals

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Theorem

Let $C$ be a contour.

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


Then:

$\displaystyle \size {\int_C \map f z \rd z} \le \max_{z \mathop \in \Img C} \size {\map f z} \map L C$

where $\map L C$ 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 $\gamma_i: \closedint {a_i} {b_i} \to \C$ for all $i \in \set {1, \ldots, n}$.

Then:

\(\displaystyle \size {\int_C \map f z \rd z}\) \(=\) \(\displaystyle \size {\sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \map f {\map {\gamma_i} t} \map {\gamma_i'} t \rd t}\) Definition of Complex Contour Integral
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \size {\int_{a_i}^{b_i} \map f {\map {\gamma_i} t} \map {\gamma_i'} t \rd t}\) Triangle Inequality for Complex Numbers
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \int_{a_i}^{b_i} \size {\map f {\map {\gamma_i} t} } \size {\map {\gamma_i'} t} \rd t\) Modulus of Complex Integral
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \max_{t \mathop \in \closedint {a_i} {b_i} } \size {\map f {\map {\gamma_i} t} } \int_{a_i}^{b_i} \size {\map {\gamma_i'} t} \rd t\) Linear Combination of Integrals
\(\displaystyle \) \(\le\) \(\displaystyle \sum_{i \mathop = 1}^n \max_{z \mathop \in \Img C} \size {\map f z} \int_{a_i}^{b_i} \size {\map {\gamma_i'} t} \rd t\) as $\map {\gamma_i} t \in \Img C$
\(\displaystyle \) \(=\) \(\displaystyle \max_{z \mathop \in \Img C} \size {\map f z} \map L C\) Definition of Length of Contour (Complex Plane)

$\blacksquare$


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