I need super help on a complex analysis problem.

[dt666999]

Homework Statement

Is there a Laurent Series for Log(z) in the Annulus 0<|z|<1?

Homework Equations

Go here for the Theorem. It is theorem 7.8:

www.math.fullerton.edu/mathews/c2003/LaurentSeriesMod.html[/URL]

(copy and paste the link below if you are having problems. Exclude the "[url]" in the link below):

[PLAIN]http://math.fullerton.edu/mathews/c2003/LaurentSeriesMod.html[url]

This whole section actually comes right out of my current textbook!

The Attempt at a Solution

From what is stated in the theorem I'm assuming yes because Log(z) is analytic for all z not equal to 0. So for any Annulus A(0,0,R), where R >0, a laurent series should exist.

I calculated a few coefficients out, but I'm not sure it is right for all n. I tried to google this situation to verify that my logic is right, but the only series I'm ever given is the maclaurin series of Log(1+z) which converges in the disk |z|<1.

Any help on this would be most apperciated it.

THanx.

 

[Hurkyl]

Log(z) is analytic for all z not equal to 0.

No it isn't. It's not even continuous on the set of all nonzero z.

 

[dt666999]

No it isn't. It's not even continuous on the set of all nonzero z.

I forgot I was looking at the principal branch LOG function, which is not analytic on the negative real axis. Thanx anyways.