- Estimation lemma
In mathematics, the estimation lemma gives an
upper bound for acontour integral . If "f" is a complex-valued,continuous function on the contour and if its absolute value |"f"("z")| is bounded by a constant "M" for all "z" on , then:where is the
arc length of . In particular, we may take the maximum:
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 |"f"("z")| for each segment. Out of all the maximum |"f"("z")|'s for the segments, there will be an overall largest one. Hence, if the overall largest |"f"("z")| is summed over the entire path then the integral of "f"("z") over the path must be less than or equal to it.
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 goes to infinity. An example of such a case is shown in the example below.Example
Problem.Find an upper bound for
:
where is the upper half-
circle withradius 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 "a", hence:Next we seek an upper bound "M" for the integrand when . By thetriangle inequality we see that:therefore: because on . Hence:Therefore we apply the estimation lemma with "M" = 1 / ("a"2 − 1)2. The resulting bound is:References
* Saff, E.B, Snider, A.D., "Fundamentals of Complex Analysis for Mathematics, Science, and Engineering" (Prentice Hall, 1993).
* Howie J.M., "Complex Analysis" (Springer, 2003).
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