Answer to Question #91403 in Complex Analysis for Sajid

Question #91403
Choose the correct answer.
Q. The residue of f(z)=sinh⁡z/z^6 is
a. 1/6!
b. 1/4!
c. 0
d. 1
1
Expert's answer
2019-07-15T11:09:01-0400


f(z)=sinh(z)/z6

Since, as per the expansion by Maclauram theorem,

sinh(z)=z+z3/3!+z5/5!+-----------

Thus,

f(z)=1/z5+1/(z33!)+1/(z15!)+---------

We need to take only the coefficient of 1/z1 for the residue. Other terms of the expansion do not have any meaning in this case;

Z0=0 is the pole of order 5 and the terms containing 1/z1 is

1/5!, which is the residue of the function.

All options are wrong.


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