# Complex Integral of $(\exp(z)-\exp(-z))/z^n$

How should I evaluate the following integral: $$\int_\gamma \frac{e^z-e^{-z}}{z^n}dz,$$where $\gamma$ is the unit circle and $n\in\Bbb N$.

My work on the integral has been to parametrize the unit circle, so I get the integral to $$\int_0^{2\pi}\frac{e^{e^{i\theta}}-e^{-e^{i\theta}}}{e^{in\theta}}ie^{i\theta}\,d\theta=i\int_0^{2\pi}\frac{e^{e^{i\theta}}-e^{-e^{i\theta}}}{e^{i(n-1)\theta}}\,d\theta.$$I got to this point, and I wasn't sure how to finish evaluating the integral.

Will someone provide only a hint?

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Hint: the integrand has a pole at $z=0$. What is the Laurent series?
@Clayton, if you are two chapters into that book, you probably know Cauchy's theorem already. Now write $e^z = 1 + z + \cdots + z^n/n! + g(z)$, where $g(z)$ is the tail. (Note that $g(z)/z^n$ is holomorphic). Do the same thing for $e^{-z}$, what can you conclude? – user27126 Feb 8 '13 at 4:59