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)=sinzz4f(z) =\frac{sinz} {z^4} has a pole of order 3 at z=0z=0



f(z)=sinzz4=zz33!+z55!...z4=f(z) =\frac{sinz} {z^4}=\frac{z-\frac{z^3}{3!}+\frac{z^5}{5!}-...}{z^4}=

=1z23!+z45!...z3=z3(1z23!+z45!...)=z3ψ(z),=\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 ψ(z)=1z23!+z45!...\psi(z) =1-\frac{z^2}{3!}+\frac{z^4}{5!}-... is holomorphic function at the point z=0z=0


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