Question

The value of the integral \[ \int_C \frac{2e^z}{(z-4)(z-2)} \, dz \] where \( C: |z| = 3 \) is

• A. \( -\pi i e^2 \)
• B. \( -2 \pi i e^2 \)
• C. \( 2 \pi i e^2 \)
• D. \( \pi i e^2 \)

Hint:

Use Cauchy's integral formula for evaluating integrals over closed contours — only consider singularities inside the contour.

Answer 1

The Correct Option is B

By Cauchy’s integral formula: \[ \int_C \frac{f(z)}{(z-a)} \, dz = 2 \pi i \, f(a) \] Now within \( |z| = 3 \), only singularity at \( z = 2 \) lies inside. So, \[ = 2 \pi i \times \frac{2 e^2}{(2-4)} \] \[ = 2 \pi i \times \frac{2 e^2}{-2} \] \[ = -2 \pi i e^2 \]