I have an integral $\oint \frac{2z}{z^2-9}dz$. It's said that the integral taken around the circle |z|=2 Here, the roots of z are $3,-3$ so it's outside the region. Does it mean the integral is 0?
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yes, because the integrand is holomorphic inside the circle of radius $2$. – peek-a-boo Mar 25 '21 at 21:11
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it's a ring, not a circle and said "around". Eventhough, is still $0$ right? – Aegean Mar 25 '21 at 21:14
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yes i meant the integrand is holonorphic in the (simply connected) open disc $U={z, :, |z|<3} $, and the ring/circle (however you want to call it) of radius 2 lies inside $U$ therefore the integral is 0 – peek-a-boo Mar 25 '21 at 22:03