Answer to Question #91401 in Complex Analysis for Sajid

Question #91401
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
1
Expert's answer
2019-07-11T08:58:02-0400

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"


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