estimating theorem of contour integral

Theorem.

If $f$ is a continuous complex function on the rectifiable curve $\gamma$ of the complex plane, then

$\left|{\displaystyle {\int }_{\gamma }}f(z)𝑑z\right|\leqq \underset{z\in \gamma }{\max }|f(z)|\cdot l,$(1)

where

$$l={\int }_{\gamma }|dz|$$

is the of $\gamma$.

The form of (1) concerning the continuous real function $f$ on the interval $[a,b]$ is

$$\left|{\int }_a^{b}f(x)𝑑x\right|\leqq \underset{a\leqq x\leqq b}{\max }|f(x)|\cdot (b-a).$$

For applications of this important theorem, see the example of using residue theorem.

Title	estimating theorem of contour integral
Canonical name	EstimatingTheoremOfContourIntegral
Date of creation	2013-03-22 15:19:36
Last modified on	2013-03-22 15:19:36
Owner	pahio (2872)
Last modified by	pahio (2872)
Numerical id	7
Author	pahio (2872)
Entry type	Theorem
Classification	msc 30E20
Classification	msc 30A99
Synonym	estimation theorem of integral
Synonym	integral estimating theorem
Related topic	MinimalAndMaximalNumber
Related topic	AnalyticContinuationOfRiemannZetaUsingIntegral
Related topic	IntegralMeanValueTheorem