Question

For the funciton $\frac{sinz}{z^3}$ of a complex variabl $z$, the point $z=0$ is

• A. a pole of order $3$
• B. a pole of order $2$
• C. a pole of order $1$
• D. not a singularity

Answer 1

The correct option is B a pole of order $2$

$f\left(z\right)=\frac{sinz}{z^3}=\frac{1}{z^3}[z-\frac{z^3}{3!}+\frac{z^5}{5!}-\frac{z^7}{7!}+....]$
(Using Meclaurin series expansion)
$=\frac{1}{z^2}-\frac{1}{3!}+\frac{z^2}{5!}-\frac{z^4}{7!}+.....$
$\therefore z=0$ is a pole of oder $2$.