Complex Analysis: Theory and Applications · Problem 6.2.10

Complex Integration — Howell & Matthews

Mathematics — Complex Analysis · ml-inequality

Problem

Use the techniques of Example 6.11 to show that (a)

\left|\displaystyle\int_C \frac{1}{z^2-1}\,dz\right| \leq \dfrac{\pi}{3}

, where

C

is the first quadrant portion of

C_2^+(0)

. (b)

\left|\displaystyle\int_{C_R^+(0)} \frac{\operatorname{Log}(z)}{z^2}\,dz\right| \leq 2\pi\left(\dfrac{\pi + \ln R}{R}\right)