what is the definition of analytic completion?
Show by example that a harmonic function need not have an analytic completion in a multiply connected domain. [HINT: Consider ln(|z|), z a complex number]
well I considered u=ln(x^2+y^2) where z=x+iy
and I figured out it was harmonic (second partial derivitives are 0) but I do not know where to go from there.
So let u=ln|z|, u is hormonic in the domain of punctured plane .
Since an analytic function is determined in any open sub-set of the plane the analytic completion is unique( up to a constant), if it exists.
So the only possible analytic completion for u would be f=lnz=u + i argz
While f can only be defined on a plane where a half line is cut.