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: Problem Overview:
This problem requires evaluating a complex contour integral. The key is to examine the integrand and its singularities' locations relative to the integration contour.

Step 2: Method:
Cauchy's Integral Theorem is used. This theorem states that if \( g(z) \) is analytic within 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: Solution:
The integrand is \( g(z) = \frac{z^3-z}{(z-z_0)^3} \). The numerator, \( z^3-z \), is a polynomial, thus analytic everywhere. The denominator, \( (z-z_0)^3 \), is zero only at \( z = z_0 \). Therefore, \( g(z) \) has a pole of order 3 at \( z = z_0 \). The problem states that \( z_0 \) is outside the closed curve C. This means \( g(z) \) is analytic within and on contour C. Therefore, by Cauchy's Integral Theorem, the integral equals zero.

Step 4: Answer:
The integral's value is 0.