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For which simple closed curves $\gamma$ is $\displaystyle\int_{\gamma} z^{2}+z+1\, dz=0$

Could someone help me through this problem?

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Do you know Cauchy's theorem? – froggie May 10 '12 at 20:44
$z^2+z+1$ has an antiderivative in the entire complex plane.... – N. S. May 10 '12 at 20:44
If and consider also the theorem of Jordan curve – Daniela del Carmen May 10 '12 at 20:48
What do you mean by "If and consider"? – GEdgar May 10 '12 at 21:05
up vote 1 down vote accepted

Every polynomial has a primitive and so you can use the fundamental theorem of calculus to conclude that the integral of a polynomial around a closed curve is zero.

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If $f(z)$ is holomorphic, then for any closed curve $\gamma$, we have $$\int_\gamma f(z)= 0$$ As other also commented: See cauchy theorem

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Also: any polynomial $f(z)$ is analytic -- or should we say holomorphic! – bgins May 10 '12 at 21:58

Yet anouther way to see this is to note that the polynomial is entire. Since it is entire, we have that, by the deformation invariance theorem, the integral is zero (since the loop can be deformed continuously to a point in the domain of analyticicity).

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