Find the residue of the function

at z=0.

Solution:



where



or




Question 2:

Find all the poles of the function

and for each one state its order.

Solution:

Poles at z=0, z=-1. At z=0 numerator vanishes to order 2, while denominator vanishes to order 3, hence the pole is of order 1. At z=-1 the numerator doesn't vanish, while denominator vanishes to order 3. Hence the pole is of order 3.


Question 3:

Find


Hint: Taylor series work better than L'Hopital's rule when applied to quotients of analytic functions.

Solution: