Cauchy integral theorem question

[JohnSimpson]

I recently did a problem in which the integral around a contour contained two residues, the sum of which was zero, so the total integral around the entire path was zero?

By the CIT, the function should then be analytic (holomorphic, if you like) inside that contour, but it isn't obviously since there are poles.

Why doesen't the CIT apply? Is the region no longer simply connected or something?

 

[morphism]

Why would the CIT imply that the function is analytic?

It looks to me that you're trying to use Morera's theorem, but incorrectly.

 

[JohnSimpson]

the cauchy riemann relations are used in the derivation (that I have) of the cauchy integral formula. unless I'm really missing something, this means that if the cauchy integral theorem is satisfied, in that the line integral around a simply closed curve C in a simply connected region is zero, then the function should be analytic within that region.

?

 

[adriank]

The region is not simply connected: it has two holes at the poles. (Rhyme unintentional.)

 

[Kaan]

I think you are just confused about the Cauchy-Goursat Theorem, it is a one sided implication which states that the integral of an analytic function over a closed contour is 0, but the converse is not generally true.

You actually don't need a simply connected region to apply Morera's Theorem. However, you need to get zero for every integral around a closed contour to prove that the function is analytic . So if you integrate over just one of the residues, you probably won't get zero, hence the function will fail to be analytic