What is the nature of the singularity at z=0 for the function f(z)=1/cos(z)+1/z?

[Winzer]

Homework Statement

Determine the nature of the singularity at z=0

Homework Equations

f(z)=\frac{1}{cos(z)}+\frac{1}{z}

The Attempt at a Solution

by expanding into series:

f(z)=\Sigma_{n=0}^{\infty} \frac{(2n)! (-1)^n}{x^{2n}} + \Sigma_{n=0}^{\infty} (-1)^n (z-1)^n
Now \frac{1}{z} has no principle part, b_m=0[/itex].<br /> This leaves the only principle part from cos. b_m=(2m)! (-1)^m. There are infinite bm<br /> so the behaviour is an essential singularity. <br /> <br /> I don&#039;t feel too confident about this answer. I feel I have overlooked a step.

 

[Dick]

Ouch. If you are looking for a singularity at z=0, why are you expanding 1/z around z=1? And cos(0)=1, it's not a singularity of 1/cos(z) at all. And you can't invert a power series by inverting each term in the power series. 1/(a+b) is not equal to 1/a+1/b.