Find the residue in $\displaystyle z=0$ of $\displaystyle f(z)=exp\left ( z+ \frac{1}{z} \right )$.

My attempt : I notice that $\displaystyle f$ is analytic except in $\displaystyle 0$. (well I think so! I guess I should show that $\displaystyle f$ is analytic. Is that sufficient that f satisfy C-R equations? Because as you'll see, I need the analyticity of $\displaystyle f$ for the coming part)

I believe that $\displaystyle \int _{\gamma} f(z)dz = 2\pi i= Res f(0)$.

I forgot something very important : why does $\displaystyle \int _{\gamma} f(z)dz = 2\pi i$? I mean, I think there's a theorem that justify it (it seems it's not Cauchy-Goursat theorem) but I forgot which one.

Am I right?