# On Analytic Continuations of complex-valued functions

1. Oct 22, 2009

### logicalmoron

Hey Folks, first post here,

I'm having significant difficulty understanding how to create an analytic continuation of a function. The topic seems straightforward (please stop me if I am wrong): if you have two functions whose laurent expansions have a radius of convergence > 0, then the functions must be equal on the domain where those radii of convergence overlap (this is likely a gross oversimplification as the topic was just introduced yesterday.)

My question is as to how you actually construct an analytic continuation of a function — say the log(z) function, with a branch cut taken from (-∞,0), to find an analytic continuation of the function on that branch cut minus the pole at z = 0.

Again this is still probably a gross oversimplification so I would really appreciate any advice/suggestions you guys have.

2. Oct 26, 2009

### skook

Log is analytic at zero when it is represented by a Taylor series about -1.