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QUESTION


Let $C$ be the arc of the circle $|z|=2$ from $z=2$ to $z=2i$ that
lies in the first quadrant. Without evaluating the integral,
 show that
$$\Big|\int_C{{dz\over z^2-1}}\Big|\le {\pi\over 3}.$$



ANSWER


Length of contour $=4\pi/4=\pi$. Also (looking at modulus of
integrand) $$|{1\over z^2-1}|\le {1\over |z|^2-1}\le {1\over 3}.$$
(We have used the backward triangle inequality here.) Thus by the
Estimation Theorem $$|\int_C{dz\over z^2-1}|\le \pi/3.$$




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