Question

The value of the integral \( \int_C \frac{6z - 5}{z^2 + 4z + 5} \, dz \), where \( C \) is the circle \( |z| = 1 \), is _______

• A. 0
• B. \(2\pi i\)
• C. \(\pi\)
• D. \(\frac{\pi i}{2}\)

Hint:

If a function is analytic inside and on a closed contour, the integral over the contour is zero (Cauchy’s Theorem).

Answer 1

The Correct Option is A

We are integrating a complex function over a closed contour \(C\), where \(C: |z| = 1\). Let's first factor the denominator: \[ z^2 + 4z + 5 = (z + 2)^2 + 1 \Rightarrow \text{roots: } z = -2 \pm i \] Both singularities lie outside the contour \(|z| = 1\).
By Cauchy's theorem, since the integrand is analytic inside and on the closed curve \(C\), the integral is zero. \[ \oint_C f(z)\, dz = 0 \] Final Answer: (1) 0