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Estimation Lemma


The estimation lemma, also called the ML inequality (short for "maximum times length"), bounds a contour integral by the maximum magnitude of its integrand times the arc length of the contour. If f is a continuous function with complex values on a rectifiable contour gamma, |f(z)|<=M on gamma, and L is the arc length of gamma, then

 |int_gammaf(z)dz|<=ML.

In particular, one may take M=max_(z in gamma)|f(z)|.


See also

Arc Length, Contour, Contour Integral, Triangle Inequality

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References

Saff, E. B. and Snider, A. D. Fundamentals of Complex Analysis for Mathematics, Science, and Engineering, 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 1993.

Cite this as:

Weisstein, Eric W. "Estimation Lemma." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EstimationLemma.html

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