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

1. Mar 12, 2009

### opticaltempest

1. The problem statement, all variables and given/known data

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.

$$\oint_{C}^{}{\frac{z^2+1}{e^{iz}-1}}$$

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

3. 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

$$\frac{f(z)}{z}$$,

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?

Last edited by a moderator: May 4, 2017
2. Mar 12, 2009

### lurflurf

3. Mar 12, 2009

### opticaltempest

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. Mar 12, 2009

### lurflurf

The residue thorem is a simple use of the integral formula.
write f(z)=[z*f(z)]/z

5. Mar 13, 2009

### latentcorpse

so are these the steps:

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

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