Is Cauchy's integral formula applicable to this type of integral?

In summary, the conversation discusses the use of Cauchy's integral formula on a specific integral with a unit circle contour, and the realization of a pole at z=0. It is determined that the residue theorem can be used instead and the steps for using it are outlined, including a Laurent expansion and finding the residue coefficient.
  • #1
opticaltempest
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Homework Statement



I am trying to determine if Cauchy's integral formula will work on the following integral, where the contour C is the unit circle traversed in the counterclockwise direction.

[tex]\oint_{C}^{}{\frac{z^2+1}{e^{iz}-1}}[/tex]


Homework Equations


See Cauchy's Integral Formula - http://en.wikipedia.org/wiki/Cauchy_integral_formula" [Broken]

The Attempt at a Solution



I realize that there is a pole at z=0. I realize that if I could get this integral into the form

[tex]\frac{f(z)}{z}[/tex],

with f(z) being analytic in and on the contour C, then I could use the formula. However, I'm not sure how to get the integrand in that form. Is it even possible to use Cauchy's integral formula on this integral, or do I need to use a different method to evaluate this integral?
 
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  • #3
Ok, so it appears that I need to use the residue theorem in order to evaluate this integral. I was hoping I could just use the integral formula. I haven't got to study the residue theorem yet in my text. Thanks
 
  • #4
The residue thorem is a simple use of the integral formula.
write f(z)=[z*f(z)]/z
 
  • #5
so are these the steps:

do a laurent expansion of denominator
cancel with stuff in the numerator
then the coefficient of the [itex]z^{-1}[/itex] term gives us the residue
multiply this by [itex]2 \pi i[/itex] to give the integral's value

im not too sure about the first of those two steps?
 

1. What is Cauchy's integral formula?

Cauchy's integral formula is a result in complex analysis that relates the values of a function inside a closed contour to the values of the function on the contour itself. It is often used to evaluate complex integrals and calculate residues of functions.

2. How is Cauchy's integral formula applicable to integrals?

Cauchy's integral formula can be used to evaluate integrals in complex analysis, especially those involving functions with singularities. It allows us to express the value of an integral in terms of the values of the function on the contour of integration, making it a powerful tool for computing complex integrals.

3. Is Cauchy's integral formula applicable to all types of integrals?

No, Cauchy's integral formula is applicable only to integrals in complex analysis. It cannot be used for real integrals or for integrals involving only real variables. It also has certain conditions that must be met, such as the function being analytic within the contour of integration.

4. Can Cauchy's integral formula be used for integrals with multiple variables?

No, Cauchy's integral formula is applicable only to integrals of functions with a single complex variable. It cannot be used for integrals involving multiple variables, as it is a result in complex analysis.

5. Are there any limitations to using Cauchy's integral formula for evaluating integrals?

Yes, there are some limitations to using Cauchy's integral formula. As mentioned earlier, the function must be analytic within the contour of integration. Additionally, the contour must be simple and closed, and the function must be continuous on the contour and its interior. Violation of any of these conditions may result in incorrect or undefined values for the integral.

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