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