Estimation lemma

In mathematics the estimation lemma, also known as the ML inequality, gives an upper bound for a contour integral.

Intuitively, the lemma is very simple to understand.

Formally, the inequality can be shown to hold using the definition of contour integral, the absolute value inequality for integrals and the formula for the length of a curve as follows: The estimation lemma is most commonly used as part of the methods of contour integration with the intent to show that the integral over part of a contour goes to zero as |z| goes to infinity.

Find an upper bound for where Γ is the upper half-circle |z| = a with radius a > 1 traversed once in the counterclockwise direction.

First observe that the length of the path of integration is half the circumference of a circle with radius a, hence Next we seek an upper bound M for the integrand when |z| = a.