ML estimate

The ML estimate is a bound on the magnitude of a contour integral:

For a contour $C$ of length $L$ and a continuous function $f$ with $|f(z)|\le M$ on $C$,

$\left|{\int }_{C}f(z)dz\right|\le M\cdot L.$

Hence the name: Maximum of $|f|$ times Length of $C$.

Proof

$\left|{\int }_{C}f(z)dz\right|=\left|{\int }_a^{b}f(z(t))z^{\prime }(t)dt\right|\le {\int }_a^b|f(z(t))||z^{\prime }(t)|dt\le M{\int }_a^b|z^{\prime }(t)|dt=M\cdot L.$

First step: triangle inequality for integrals. Second: bound the integrand. Third: $\int |z^{\prime }(t)|dt$ is the Arc length of the curve, which is $L$. ∎

Standard use: showing semicircle contributions vanish

The most common application: estimating the contribution of a large semicircle in calculations of real improper integrals via contour integration.

Example. Let $C_R$ be the upper semicircle $|z|=R$, $\mathrm{Im}(z)\ge 0$. Estimate $\left|{\int }_{C_R}\frac{dz}{z^2+1}\right|$ as $R\to \infty$.

Length $L=\pi R$.

On $C_R$, $|z|=R$. By the reverse Triangle inequality, $|z^2+1|\ge |z{|}^2-1=R^2-1$, so

$\left|\frac{1}{z^2+1}\right|\le \frac{1}{R^2-1}.$

Apply ML:

$\left|{\int }_{C_R}\frac{dz}{z^2+1}\right|\le \frac{\pi R}{R^2-1}\to 0\text{ as }R\to \infty .$

So the semicircle contribution vanishes. This is the standard “vanishing semicircle” estimate used repeatedly in Chapter 14 of Vector Calculus and Complex Analysis to compute real improper integrals via contour integration. The trick: close a real integral ${\int }_{-\infty }^{\infty }f(x)dx$ with a large semicircle in the upper half-plane; the semicircle contribution dies by ML, and the real integral equals $2\pi i\sum$(residues) by the Residue theorem.

When ML is sharp

ML gives an upper bound, not the exact value. It’s sharp (equality) only when $|f(z)|$ is constant on $C$ and $z^{\prime }(t)$ has constant phase relative to $f$ — rare in practice. In typical use it overestimates by a constant factor, but the estimate is more than good enough for arguing limits.

Jordan’s lemma (sharpening)

For integrands of the form $f(z)e^{iaz}$ with $a>0$ and $|f(z)|\to 0$ as $|z|\to \infty$ in the upper half-plane, the semicircle contribution dies even when ML alone wouldn’t suffice — the exponential decay $|e^{iaz}|=e^{-ay}$ for $y>0$ helps. This is Jordan’s lemma, a finer-grained version of ML used for $\int f(x)\cos (ax)dx$-type integrals.

In context

The ML estimate is the standard tool for:

• Bounding contour integrals when exact computation is hard or infeasible.
• Proving that semicircle / quarter-circle / rectangular-corner contributions vanish in limits.
• Establishing convergence of Power series (the proof of the radius of convergence uses an ML-style bound on Taylor coefficients via Cauchy’s estimate).
• The proof of the Cauchy integral formula (showing the error term in deforming a contour to a small circle vanishes).

It is the contour-integral analog of the inequality $\left|{\int }_a^{b}f(x)dx\right|\le M(b-a)$ from real analysis — just dressed up for complex paths.