Estimation lemma

Statement

Let $\gamma : [a, b] \to \mathbb{C}$ be a continuous and piecewise continuously differentiable path with length $L(\gamma) = \int_a^b |\gamma'(t)| \, dt$. Let $f$ be a complex function that is continuous on the image of $\gamma$. Let $M \ge 0$ be a real number with $|f(z)| \le M$ for each $z$ on the image of $\gamma$. Then the contour integral of $f$ along $\gamma$ satisfies $$\left| \int_\gamma f(z) \, dz \right| \le M \, L(\gamma).$$

Remarks

For a circle integral over $|z - c| = R$, the bound is $\left| \oint f(z) \, dz \right| \le 2 \pi R M$. Proof: $\left| \int_a^b g(t) \, dt \right| \le \int_a^b |g(t)| \, dt$ for a complex function $g$, and $|f(\gamma(t)) \gamma'(t)| \le M |\gamma'(t)|$. A typical use shows that the integral over a large half-circle goes to $0$ as the radius goes to $\infty$.

Explanations

L1 human

The integral cannot be larger than the largest size of the function times the length of the path. For example, if $|f| \le 1$ on a circle of radius $3$, then the integral over the circle has size at most $6\pi$.