Cauchy integral theorem question

In summary, the conversation discusses the application of the Cauchy Integral Theorem and Morera's Theorem in determining the analyticity of a function within a contour. While the CIT implies that the function is analytic in a simply connected region, it may not hold true in a region with holes or poles. Additionally, the one-sided implication of Morera's Theorem requires the integral to be zero for every contour to prove analyticity, regardless of the region's connectivity.
  • #1
JohnSimpson
92
0
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?
 
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  • #2
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.
 
  • #3
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.

?
 
  • #4
The region is not simply connected: it has two holes at the poles. (Rhyme unintentional.)
 
  • #5
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
 

FAQ: Cauchy integral theorem question

What is the Cauchy integral theorem?

The Cauchy integral theorem, also known as Cauchy's integral formula, is a fundamental theorem in complex analysis that states that if a function is analytic in a simply connected region, then its integral along any closed path in that region is equal to the sum of its values at all points inside the path.

How is the Cauchy integral theorem used in mathematics?

The Cauchy integral theorem is used in mathematics to evaluate complex integrals, to prove the existence of analytic functions, and to establish the behavior of analytic functions near singularities. It is also a key tool in complex analysis and has applications in physics and engineering.

What is the difference between the Cauchy integral theorem and the Cauchy residue theorem?

The Cauchy integral theorem and the Cauchy residue theorem are closely related but have different purposes. The Cauchy integral theorem is used to evaluate complex integrals, while the Cauchy residue theorem is used to find the value of an analytic function at a point inside a closed contour by calculating the residues of the function at its singularities.

How is the Cauchy integral theorem related to Green's theorem?

The Cauchy integral theorem is a special case of Green's theorem, which is a fundamental theorem in vector calculus. Green's theorem relates double integrals over a region in the plane to line integrals around the boundary of that region. The Cauchy integral theorem can be derived from Green's theorem by taking the limit as the region shrinks to a point.

What are the limitations of the Cauchy integral theorem?

The Cauchy integral theorem is limited to analytic functions, which have a derivative at every point in their domain. It also only applies to simply connected regions, meaning that there are no holes or gaps in the region. Additionally, it only applies to closed paths, so it cannot be used to evaluate integrals along open curves.

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