The ML Inequality and Estimates

The ML inequality is the mechanism behind vanishing-arc arguments used to evaluate real integrals by residues: closing a contour with a large semicircular arc and showing its contribution vanishes as the radius grows is a direct application, foreshadowing Evaluation of Real Integrals via the residue theorem. It underlies the proof of Cauchy's Integral Formula and the closely related Cauchy estimates on derivatives of holomorphic functions, which in turn power Liouville's theorem and the fundamental theorem of algebra. In applied contexts, the same style of bound governs stability estimates in the study of Laplace and Fourier transform inversion contours in signal processing and control theory, where engineers must show that "tail" contributions to an inversion integral vanish. In numerical analysis and scientific computing, ML-type bounds justify truncating or deforming contours in numerical contour-integration schemes (such as those used to compute matrix functions or solve certain PDEs spectrally) while controlling the resulting error. In physics, similar length–modulus estimates appear in scattering theory and quantum field theory when contour-deformation arguments (Wick rotation, dispersion relations) require showing arc contributions at infinity vanish.

Ask the tutor: How do I choose the right auxiliary contour and correctly estimate M on it when using the ML inequality to evaluate a specific real integral by residues?

Think of the contour integral $\int_\gamma f(z)\,dz$ as accumulated "signed area/rotation" swept out as a particle traces $\gamma$, weighted at each instant by $f(\gamma(t))\gamma'(t)$. The absolute value of an accumulated sum of vectors can never exceed the sum of their individual lengths — this is just the triangle inequality applied repeatedly. In the continuum limit that defines the integral, "sum of individual lengths" becomes $\int_a^b |f(\gamma(t))|\,|\gamma'(t)|\,dt$, and since $|f(\gamma(t))| \le M$ throughout, that quantity is at most $M$ times $\int_a^b |\gamma'(t)|\,dt = L$. Geometrically, picture chopping the contour into many tiny pieces. Each piece contributes a small complex number to the Riemann sum approximating the integral, and the magnitude of that contribution is roughly $|f(z_k^*)| \cdot |\Delta z_k|$ — the size of $f$ there, times the length of that little piece of arc. No matter how these small complex vectors point (they can point in wildly different directions and even cancel), the sum of their magnitudes is an upper bound for the magnitude of their sum. Bounding $|f|$ uniformly by $M$ and summing the arc-length pieces to get $L$ gives the crude but ironclad bound $ML$. This is why the inequality is so often used with vanishing arcs: if you can show $M \to 0$ faster than $L$ grows (or $L \to 0$ while $M$ stays bounded), the whole integral is forced to zero, without ever computing it.

Example 1 — a genuinely curved, non-circular, non-closed contour. Let $\gamma$ be the straight segment from $z_1 = 1+i$ to $z_2 = 4 + 5i$, and let

We bound $\left|\int_\gamma f(z)\,dz\right|$ without evaluating the integral. Length. $\gamma$ is parametrized by $\gamma(t) = (1+i) + t\big[(4+5i)-(1+i)\big] = (1+i) + t(3+4i)$, $t \in [0,1]$, so $\gamma'(t) = 3+4i$, a constant, and

(Indeed $L$ equals the straight-line distance $|z_2 - z_1| = |3+4i| = 5$, as it must for a segment.) Bound on $|f|$. We need $M$ such that $|f(z)| \le M$ for $z$ on the segment. First bound the numerator: for $z = x+iy$ on the segment, $x$ ranges over $[1,4]$ and $y$ over $[1,5]$ (both monotonic in $t$), so

using that $|z(t)|$ is a convex-combination-controlled quantity maximized at an endpoint for a line segment not passing further from the origin than its endpoints in this case — more simply and rigorously, for any $z$ on the segment, $z = (1-t)z_1 + t z_2$, so by the triangle inequality

Hence $|z^2| \le 41$ and $|z^2+2| \le 41 + 2 = 43$. Now bound $|z - 2i|$ from below. The point $2i$ is at distance

from the segment. Compute the foot of the perpendicular: the line through $\gamma$ has direction $3+4i$ (unit vector $\frac{1}{5}(3+4i)$) and passes through $1+i$. The vector from $1+i$ to $2i$ is $-1+i$. Its projection length onto the direction is

(using the real inner product $(a+bi)\cdot(c+di) = ac+bd$), which lies in $[0,5]$ (the parametrized arc-length range), so the foot of the perpendicular lands on the segment, at distance

along the parametrization $t\in[0,1]$. The perpendicular distance is

So the minimum of $|z-2i|$ over the segment is $d = 7/5$, hence

Therefore, for $z \in \gamma^*$,

Take $M = \dfrac{215}{7}$. Conclusion. By the ML inequality,

This bound was obtained with no anti-derivative and no parametrized computation of the integral itself — only geometry of the segment and elementary modulus estimates.

Example 2 — a large circular arc used to force a limit to zero (foreshadowing residue-style arguments). Let $\Gamma_R$ be the arc of the circle $|z| = R$ (with $R > 3$) lying in the upper half-plane, traversed from $R$ to $-R$ counterclockwise (i.e. $z = Re^{i\theta}$, $\theta \in [0,\pi]$), and let

We show $\displaystyle \int_{\Gamma_R} f(z)\, dz \to 0$ as $R \to \infty$. Length. $\Gamma_R$ is parametrized by $\gamma(\theta) = Re^{i\theta}$, $\theta \in [0,\pi]$, so $\gamma'(\theta) = iRe^{i\theta}$, $|\gamma'(\theta)| = R$, and

Bound on $|f|$. For $|z| = R$, the reverse triangle inequality gives

which is positive once $R > 2$ (certainly true since $R>3$). Also $|z| = R$ exactly. Hence for $z \in \Gamma_R^*$,

Applying the ML inequality.

As $R \to \infty$, $\dfrac{\pi R^2}{R^3-8} = \dfrac{\pi/R}{1 - 8/R^3} \to \dfrac{0}{1} = 0$. By the squeeze theorem (the corollary above), $\int_{\Gamma_R} f(z)\, dz \to 0$ as $R \to \infty$. This exact style of estimate — bound the numerator degree against the denominator degree, multiply by the arc length $\pi R$, and let $R \to \infty$ — is the standard first step in evaluating real integrals such as $\int_{-\infty}^{\infty} \frac{x}{x^3+8}\,dx$-type problems (after suitable adjustment for convergence) by closing contours in the upper half-plane, a technique built directly on top of this estimate.