Answer to Question #91403 in Complex Analysis for Sajid

Question

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

Answer 1

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.