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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!

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up vote 1 down vote accepted

The problem is that $0$ is outside your circle $|z-3i|=1$, not inside.

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I see, so the integral is zero using the Cauchy integral theorem. – Chris Jul 23 '12 at 18:51

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