Estimation lemma

Template:Short description In complex analysis, the estimation lemma, also known as the Template:Mvar inequality, gives an upper bound for a contour integral. If Template:Mvar is a complex-valued, continuous function on the contour Γ and if its absolute value Template:Abs is bounded by a constant Template:Mvar for all Template:Mvar on Γ, then

$|\underset{\mathrm{\Gamma }}{\int }f(z)dz|\le Ml(\mathrm{\Gamma }),$

where l(Γ) is the arc length of Γ. In particular, we may take the maximum

$M:=\underset{z\in \mathrm{\Gamma }}{\sup }|f(z)|$

as upper bound. Intuitively, the lemma is very simple to understand. If a contour is thought of as many smaller contour segments connected together, then there will be a maximum Template:Abs for each segment. Out of all the maximum Template:Abss for the segments, there will be an overall largest one. Hence, if the overall largest Template:Abs is summed over the entire path then the integral of f (z) over the path must be less than or equal to it.

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:

$|\underset{\mathrm{\Gamma }}{\int }f(z)dz|=|\underset{\alpha }{\overset{\beta }{\int }}f(\gamma (t)){\gamma }^{\prime }(t)dt|\le \underset{\alpha }{\overset{\beta }{\int }}|f(\gamma (t))||{\gamma }^{\prime }(t)|dt\le M\underset{\alpha }{\overset{\beta }{\int }}|{\gamma }^{\prime }(t)|dt=Ml(\mathrm{\Gamma })$

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 Template:Abs goes to infinity. An example of such a case is shown below.

Example

File:Upper halfcircle with i.svg

The contour Γ.

Problem. Find an upper bound for

$\left|\underset{\mathrm{\Gamma }}{\int }\frac{1}{(z^2+1{)}^2}dz\right|,$

where Γ is the upper half-circle Template:Abs = a with radius a > 1 traversed once in the counterclockwise direction.

Solution. First observe that the length of the path of integration is half the circumference of a circle with radius Template:Mvar, hence

$l(\mathrm{\Gamma })=\frac{1}{2}(2\pi a)=\pi a.$

Next we seek an upper bound Template:Mvar for the integrand when Template:Abs = a. By the triangle inequality we see that

$|z{|}^2=\left|z^2\right|=|z^2+1-1|\le |z^2+1|+1,$

therefore

$|z^2+1|\ge |z{|}^2-1=a^2-1>0$

because Template:Abs = a > 1 on Γ. Hence

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

Therefore, we apply the estimation lemma with M = Template:Sfrac. The resulting bound is

$\left|\underset{\mathrm{\Gamma }}{\int }\frac{1}{{(z^2+1)}^2}dz\right|\le \frac{\pi a}{{(a^2-1)}^2}.$

See also

• Jordan's lemma

References

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