Question

The value of the integral:

\[ \oint_C \frac{z^3 - z}{(z - z_0)^3} \, dz \] where \( z_0 \) is outside the closed curve \( C \) described in the positive sense, is:

• A. 1
• B. 0
• C. \( -\frac{8\pi i}{3}e^{-2} \)
• D. \( \frac{2\pi i}{3}e^2 \)

Hint:

In any contour integral problem, the first and most crucial step is to identify the singularities of the integrand and determine their location relative to the contour. If all singularities are outside the contour, the integral is immediately zero by Cauchy's Theorem, saving you from any complex calculations with Cauchy's Integral Formula or the Residue Theorem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem involves evaluating a complex contour integral. The key is to analyze the integrand and the position of its singularities relative to the integration contour.

Step 2: Key Formula or Approach:
The problem is solved using Cauchy's Integral Theorem. The theorem states that if a function \( g(z) \) is analytic at all points inside and on a simple closed contour C, then the integral of \( g(z) \) over C is zero. \[ \oint_C g(z) dz = 0 \]
Step 3: Detailed Explanation:
The integrand is the function \( g(z) = \frac{z^3-z}{(z-z_0)^3} \). The numerator, \( z^3-z \), is a polynomial and thus is analytic everywhere in the complex plane (it is an entire function). The denominator, \( (z-z_0)^3 \), is zero only at \( z = z_0 \). Therefore, the integrand \( g(z) \) has a single singularity (a pole of order 3) at the point \( z = z_0 \). The problem statement specifies that the point \( z_0 \) is outside the closed curve C. This means that the function \( g(z) \) is analytic at all points inside and on the contour C. Therefore, by Cauchy's Integral Theorem, the integral must be zero.

Step 4: Final Answer:
The value of the integral is 0.