1. Another Cauchy integral

Another question to be done using Cauchy's integral formula:

$\oint\frac{e^z}{z(z^2-9)} \, {\color{red}dz}$ with $|z-3|=1$

Start by letting $f(z) = \frac {e^z}{z}$ since z has no zeros in $|z-3|=1$

then $\oint\frac{f(z)}{(z^2-9) - z_0}$ with $z_0 = +-3$

So $2\pi i * f(z_0) = 2\pi i*\frac{e^3}{3}$ and $2\pi i*\frac{e^{-3}}{-3}$

Did i go wrong anywhere?

2. Originally Posted by Richmond
Another similiar question,

$\oint\frac{e^z}{z(z^2-9)} \, {\color{red}dz}$ with $|z-3|=1$

Start by letting $f(z) = \frac {e^z}{z}$ since z has no zeros in $|z-3|=1$

then $\oint\frac{f(z)}{(z^2-9) - z_0}$ with $z_0 = +-3$

So $2\pi i * f(z_0) = 2\pi i*\frac{e^3}{3}$ and $2\pi i*\frac{e^{-3}}{-3}$

Did i go wrong anywhere?
The contour encloses only one singularity: A simple pole at z = 3. So take $f(z) = \frac{e^z}{z(z + 3)}$.