Complex integral is zero but fn. is not analytic

[ppy]

Hi
I have been using a textbook which shows that ∫cos z/z^2 around the circle |z|=1 is zero by doing a Laurent expansion and finding the residue is zero.
I was under the impression that only analytic functions have a integral of zero around a closed surface. ( the Cauchy-Goursat Theorem ). The above function is not analytic at z=0 so can non-analytic functions have a closed integral of zero ?

Thanks

 

[Mandelbroth]

Hi
I have been using a textbook which shows that ∫cos z/z^2 around the circle |z|=1 is zero by doing a Laurent expansion and finding the residue is zero.
I was under the impression that only analytic functions have a integral of zero around a closed surface. ( the Cauchy-Goursat Theorem ). The above function is not analytic at z=0 so can non-analytic functions have a closed integral of zero ?

Thanks

Of course. Observe: $$\int\limits_{[0,2\pi)}\frac{\cos(e^{i\theta})}{e^{2i\theta}}ie^{i\theta}d\theta = i\int\limits_{[0,2\pi)}\frac{\cos(e^{i\theta})}{e^{i\theta}}d\theta=0.$$

 

[ppy]

Thanks for that. So if the closed integral is zero it doesn't prove that the function is analytic ? Does it infer anything ?
Thanks

 

[Mandelbroth]

Thanks for that. So if the closed integral is zero it doesn't prove that the function is analytic ? Does it infer anything ?
Thanks

Not really.

It implies that the contour integral is 0, though.

 

[mathwonk]

you are missing a hypothesis of morera's theorem that the integrand be continuous.

 

[Mandelbroth]

you are missing a hypothesis of morera's theorem that the integrand be continuous.

This is true.

If the integrand is continuous and the integral is 0, then, with a couple of other conditions, the integrand is holomorphic.