Suppose \(\gamma\) is a curve and \(f\) is continuous on \(\gamma\), then

\[ \begin{aligned} \left| \int_\gamma f(z) \, dz \,\right|\, \le \int_\gamma \left|\, f(z) \,\right|\, \left| \, dz \,\right| \\ \end{aligned} \]

In particular, if \(\left|\, f(z) \,\right| \le M \) on \(\gamma\), then

\[ \begin{aligned} \left| \int_\gamma f(z) \, dz \,\right|\, \le M \int_\gamma \, \left| \, dz \,\right| = M \cdot \text{length}(\gamma) \\ \end{aligned} \]

!

For example, suppose \(\gamma(t) = t + it, 0 \le t \le 1\), what is the upper bound of \(\displaystyle \int_\gamma z^2\,dz \) ? It should be easy to check that it's \(2\sqrt{2}\). Turns out a better and indeed exact estimate is \(\displaystyle \frac{2}{3}\sqrt{2}\), using the first part of the theorem !

More at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 7:36 PM