Question

Consider the integral \[ I = \frac{1}{2 \pi i} \oint \frac{1}{(z^4 - 1)(z - \frac{a}{b})(z - \frac{b}{a})} \, dz, \] where \( z \) is a complex variable and \( a, b \) are positive real numbers. The integral is taken over a unit circle with center at the origin. Which of the following option(s) is/are correct?

• A. \( I = \frac{5}{8} \) when \( a = 1, b = 2 \)
• B. \( I = \frac{10}{3} \) when \( a = 1, b = 3 \)
• C. \( I = \frac{5}{8} \) when \( a = 2, b = 1 \)
• D. \( I = \frac{5}{8} \) when \( a = 3, b = 2 \)

Hint:

For integrals involving rational functions with poles inside the unit circle, use the residue theorem to compute the contour integral.

Answer 1

The Correct Option is A, C

Step 1: The integral involves a rational function in \( z \) and is taken over a unit circle. To solve it, we can use the residue theorem to compute the integral by identifying the poles inside the unit circle. 
Step 2: The poles of the integrand occur where the denominators are zero. These are the points where \( z^4 - 1 = 0 \), and where \( z - \frac{a}{b} = 0 \) and \( z - \frac{b}{a} = 0 \). 
Step 3: After identifying the poles and applying the residue theorem, we find that the correct values for \( I \) are \( \frac{5}{8} \) when \( a = 1, b = 2 \) and when \( a = 2, b = 1 \), and the corresponding answer options are (A) and (C).