Applying Cauchy's Integral Formula for $\oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} d z$

Question

So, here is a problem:

$\oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} d z$

the answer is $2i$

I cant understand what I am doing wrong:

Cauchy's Integral Formula:

$f(a)=\frac{1}{2 \pi i} \oint_{\gamma} \frac{f(z)}{z-a} d z$

In this case:

$f(a) = \{a=2\} = \exp(\frac{1}{2-1}) = \exp(1)$

Thus

$\oint_{|z|=4} \frac{e^{1 /(z-1)}}{z-2} d z = 2 \pi i exp(1)$

???

Answer 1

You cannot apply Cauchy's integral formula here because $\exp\left(\frac1{z-1}\right)$ is not analytic on $\overline{D(0,4)}$. So, it is natural that it does no provide the correct answer. I suggest that you try the residue theorem instead.