Let γ be the positively oriented circle in the complex plane given by {z ∈ \(\mathbb{C}\) ∶ |z - 1| = 1/2}. The line integral

\(\int_\gamma \frac{z e^{1 / z}}{z^2-1} d z\) equals

Concept:

Cauchy Integral Theorem:

If a complex function f(z) is analytic within and on a closed contour C inside a simply-connected domain, and if a is any point in the middle of C, then

f(a) = \(\frac{1}{2 π i} \int_C \frac{f(z)}{z-a}\)

Explanation:

Singular points are given by 

z² - 1 = 0 ⇒ z = 1, z = -1

The only singular point that lies inside γ is z = 1.

Let f(z) = \(\frac{ze^{1/z}}{z+1}\)

Hence using Cauchy's Integral test

\(\int_\gamma \frac{z e^{1 / z}}{z^2-1} d z\) = 2πi f(1) = 2πi × (e/2) = iπe

 Option (1) is correct.