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

[opticaltempest]

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.

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

Homework Equations

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

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?

 

[lurflurf]

read futher
http://en.wikipedia.org/wiki/Residue_theorem
you will see
your integral=2pi*i[lim z->0 z/(exp(i*z)-1)(z^2+1)]

it is as simple as writing f(z) as [z*f(z)]/z

 

[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

 

[lurflurf]

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

 

[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?