Question

The contour integral ${\int }_c{e}^{1/z}dz$ with $C$ as the counter clock wise unit circle in the z-plnae is equal to

• A. $0$
• B. $2\pi$
• C. $2\pi \sqrt{-1}$
• D. $\alpha$

Answer 1

The correct option is C $2\pi \sqrt{-1}$

The Laurent Expansion in the negihbourhood of $z=0$ is given as
$f\left(z\right)=e^{\frac{1}{z}}=1+1\frac{1}{z}+\frac{1}{2!z^2}+\frac{1}{3!z^3}+....$
So, $R_1=Resf\left(z\right);(z=0)$
$=$ Coefficient of $\frac{1}{Z}$ in Laurent expansion
So by Cauchy residue theorem
$\therefore \int e^{\frac{1}{z}}dz=2\pi i\left(R_1\right)=2\pi i\left(1\right)=2\pi \sqrt{-1}$