Question

Consider the complex function \[ f(z) = \dfrac{z^2 \sin z}{(z - \pi)^4}. \] At $z = \pi$, which of the following options is(are) CORRECT?

• A. The order of the pole is 4
• B. The order of the pole is 3
• C. The residue at the pole is $\dfrac{\pi}{6}$
• D. The residue at the pole is $\dfrac{2\pi}{3}$

Hint:

When determining the order of a pole from a Laurent series, check the power of the singularity term in the denominator. Here, $(z - \pi)^4$ indicates a 4th order pole.

Answer 1

The Correct Option is A

- The function $f(z)$ has a pole at $z = \pi$, and we need to determine its order. The denominator has $(z - \pi)^4$, which means it suggests a pole of order 4 at $z = \pi$.
- Therefore, the correct answer for the order of the pole is 4.