Complex Variable Analysis: What is the order of the pole at z=1?

[GluonZ]

I'm supposed to evaluate where a function is a regular point, essential singularity, or a pole (and of what order) at a specific location.

Problem here.

Evaluated at (where else but) z=1.

I know its not a regular point since it doesn't evaluate to a simple Taylor Series... likewise -- I know its not an essential singularity since its Lorentz series doesn't go on forever (unless I'm entirely wrong).

I cannot tell what order the pole is though:

Using l'Hopital's rule (0/0) would suggest that its a pole of order 1... but applying it directly which the textbooks seem to do would suggest its a pole of order 2.

It would be easier if I could conform it to a Lorentz series but across 7 textbooks I have on the topic of Complex Analysis (not kidding -- I just bought one today -- 120$ -- only a dozen problems on the topic -- and no solutions nor even answers). "Arfken" -- good for reference -- but I really should have bought Boas.

 

[AiRAVATA]

A pole is simple if the limit

$$\lim_{z\rightarrow z_0}(z-z_0)f(z)$$

exist and is finite.

A pole is of order $n$ if the limit

$$\lim_{z\rightarrow z_0}(z-z_0)^n f(z)$$

exist and is finite.

(Why?)

---EDIT---

You should read Marsden's or Churchill's book on the topic. Another good, but more advanced reference is Ahlfors.

 

[DeadWolfe]

I would think that your pole is of order one, since your function is equal to (z+1)/(z-1)