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
Best Answer
The residue theorem is not restricted to functions with poles. A precise statement can be found in Wikipedia: Residue Theorem:
$f$ 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.