# Estimation Lemma for Contour Integrals

## 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:

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

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 $C_k$ be parameterized by the smooth path $\gamma_k: \closedint {a_k} {b_k} \to \C$ for all $k \in \set {1, \ldots, n}$.

Then:

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

$\blacksquare$

## Also known as

Some sources give this as Triangle Inequality for Contour Integrals.

Both names are used on $\mathsf{Pr} \infty \mathsf{fWiki}$ in order to enhance accessibility.