# estimating theorem of contour integral

###### Theorem.

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

 $\displaystyle\left|\int_{\gamma}f(z)\,dz\right|\leqq\max_{z\in\gamma}|f(z)|% \cdot l,$ (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

 $\left|\int_{a}^{b}f(x)\,dx\right|\leqq\max_{a\leqq x\leqq b}|f(x)|\cdot(b\!-\!% a).$

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