Is $\int_{|z|=2}\exp\bigl(e^{1/z}\bigr)\,dz = 2\pi ie$? 
Question is to find $$\int_{|z|=2}\exp\bigl(e^{1/z}\bigr)\,dz$$ 

Cauchy residue theorem says 

If $f(z)$ is regular, except at a finite number of poles within a closed contour $C$ and continuous on the boundary of $C$, then $ \int_{C} f(z)\,dz=2\pi i \sum R$, where $\sum R$ is the sum of the residues of $f(z)$ at its poles within $C$.

In the above integral $f(z)= \exp\bigl(e^{1/z}\bigr)$ has essential singularity at $z=0$.
Also residue for a function at a point $a$ is the coefficient of $\frac{1}{z}$ in its Laurent series expansion about $a$.
If we write the Laurent series expansion for $\exp\bigl(e^{1/z}\bigr)$, then the coefficient of $\frac{1}{z}$ is $e$ and therefore by Cauchy's residue theorem the value of the integral is $2\pi ie$.
But is this correct? In residue theorem we have residue at poles but here $z=0$ is an essential singularity.
Help me find my mistake. Thank you
 A: The residue theorem is not restricted to functions with poles. A precise statement can be found in Wikipedia: Residue Theorem:

Let $U$ be a simply connected open subset of the complex plane containing a finite list of points $a_1, \ldots, a_n$, and $f$ a function defined and holomorphic on $U \setminus \{a_1, ..., a_n\}$. Let $\gamma $ be a closed rectifiable curve in $U$ which does not meet any of the $a_k$. Then
  $$
\int_\gamma f(z) \, dz = 2 \pi i \sum_{k=1}^n \operatorname{I}(\gamma, a_k) \operatorname{Res}(f,a_k)
 $$ 
  where $\operatorname{I}(\gamma, a_k)$ denotes the winding number of $\gamma $ around $a_k$.

$f$ has “isolated singularities” at each $a_k$, that can be poles, essential singularities, or removable singularities.
If $\gamma $ is a simple positive-oriented closed curve then the formula simplifies to
$$
\int_\gamma f(z) \, dz = 2 \pi i \sum_{k}  \operatorname{Res}(f,a_k)
 $$ 
where the sum is now taken over all $a_k$ which are “inside” $\gamma$. 
In your case $f(z) = \exp\bigl(e^{1/z}\bigr)$ is holomorphic in $\Bbb C \setminus \{ 0 \}$, and therefore 
$$
\int_{|z|=2}\exp\bigl(e^{1/z}\bigr)\,dz =  2 \pi i   \operatorname{Res}(f, 0) = 2 \pi i e \, ,
$$
i.e. your result is correct.
