Question

Find all singularities of the function

sin1/z cos(z - π)
f(z) = -----------------------
(z-3π/2) (z+4i)2'
and determine the nature of each of these singularities (e.g. removable singularity, pole, essential singularity). Justify all of your steps.

Answer 1

The function has an essential singularity at z = 0, a pole of order 1 at z = 3\pi/2, and a pole of order 2 at z = -4i.

Step-by-step explanation:

To find the singularities of the given function f(z) = \frac{\sin(1/z) \cos(z - \pi)}{(z-\frac{3\pi}{2}) (z+4i)^2}, we have to identify the values of z for which the function is not defined or its behavior is not analytic.

• The first potential singularity comes from the term sin(1/z), where z = 0 will cause a problem since we cannot divide by zero. This is an essential singularity because the singularity cannot be removed and the behavior of the function near z = 0 is complex and not bound to a particular value.
• The second singularity is z = 3\pi/2, which is a zero of the denominator, thus it is a pole of order 1.
• The third and final set of singularities comes from the term (z+4i)^2, which suggests that z = -4i is a singularity. Since we have this term squared in the denominator, z = -4i is a pole of order 2.

Note that cos(z - \pi) does not introduce any singularities since cosine is an entire function, meaning it is analytic everywhere on the complex plane.