Answer to Question #91401 in Complex Analysis for Sajid

Question

Q.Choose the correct answer.
Q. Which of the following statement is true for f(z)=sin⁡z/z^4 ?
a. f has essential singularity at z=0.
b. f has pole of order 3 at z=0
c. f has removable singularity at z=0.
d. None of above

Answer 1

Answer: b) $f(z) =\frac{sinz} {z^4}$ has a pole of order 3 at $z=0$

$$f(z) =\frac{sinz} {z^4}=\frac{z-\frac{z^3}{3!}+\frac{z^5}{5!}-...}{z^4}=$$

$$=\frac{1-\frac{z^2}{3!}+\frac{z^4}{5!}-...}{z^3}=z^{-3}(1-\frac{z^2}{3!}+\frac{z^4}{5!}-...)=z^{-3}\psi(z),$$

where $\psi(z) =1-\frac{z^2}{3!}+\frac{z^4}{5!}-...$ is holomorphic function at the point $z=0$