Consider the following integral:
$\displaystyle \int_{|z-3i|=1} \frac{e^{z^2+z}}{z} dz$.
Why is my approach wrong?
Set $f(z)=e^{z^2+z}$, then from the Cauchy integral formula for circles, we have:
$\displaystyle f(0)=\frac{1}{2\pi i} \int_{|z-3i|=1} \frac{f(z)}{z} dz$.
So we have: $\displaystyle \int_{|z-3i|=1} \frac{e^{z^2+z}}{z} dz = 2 \pi i$.
Where is the mistake?
Thank you for your time!