integral of exp(1/z) over unit circle

Question

$f(z) = \exp(1/z)$

i'm trying to find a way to integrate $f(z)$ over the unit circle. i'm new at this and i have no clue where to start. i was trying to use the Cauchy's integral formula.

thank you all

Answer 1

Cauchy's integral formula won't help you here. But the residue of $f$ at $0$ is $1$ and therefore, by the residue theorem, your integral is equal to $2\pi i$.

Answer 2

HINT:

Note that on the unit circle, $z=e^{i\theta}$ and so, $e^{1/z}=\sum_{n=0}^\infty \frac{e^{-in\theta}}{n!}$.

Then integrate term by term.

Answer 3

$$\int_C{e^{1\over z}dz}=2\pi iRez_C(0)$$ Where $Rez_C(0)$ is residue of $e^{1\over z}$ in singularity 0 contained by unit circle $C$. Also: $$e^{1\over z}=1+{1\over 1!z}+{1\over 2!z^2}+{1\over 3!z^3}+...$$ therefore

$$Rez_C(0)=1$$ and $$\int_C{e^{1\over z}dz}=2\pi i$$