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I have the following complex line integral: $$ \int_{|z| = 2} \frac{z}{z - 3} $$

My prof said it is $0$, but did not explain. He just said that the point $3+0i$ lies outside the circle.

But the Cauchy integral theorem does not mention anything about it. Can anybody give me the proof and also mention does this happen even if the point lies outside the circle/loop and the function is not analytic inside it.

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The reason the answer is zero is because the function

$f(z)=\frac{z}{z-3}$

is analytic inside the circle. It is generally true that the integral around a simple closed curve of a function that is analytic in the interior is 0. So you are not using the Cauchy integral formula in this case.

If you are trying to calculate

$\int\frac{f(z)}{z-a}$

around some curve $\gamma$ where $f$ is not analytic Cauchy integral formula does not say anything about the integral no matter where $a$ is located.

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  • $\begingroup$ I need a small clarification on your last sentence of the answer , you are trying to tell that Cauchy Integral theorem says nothing about $\int_{\gamma} \frac{f(r)}{z-a} dz$ if $f$ is not analytic. Suppose f is not analytic at just a and a is at the center of the circle $c$. That is when we use Cauchy integral formula to calculate the integral right> $\endgroup$ – Jayanth Kumar Feb 16 '17 at 13:15
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By Cauchy Integral formula, if the function is analytic,it is zero.If your function is not analytics at z=3,but it lies outside.But it is analytic Inside C.

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  • $\begingroup$ I have already accepted an answer. Thanks anyways for explaining. $\endgroup$ – sashas Mar 18 '15 at 14:52

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