1. Harmonic Functions

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.

2. what is the definition of analytic completion?

3. Originally Posted by xxp9 what is the definition of analytic completion?
Analytic completion for a function u is when there exists a harmonic function v in a simply connected domain such that u+iv is analytic.

4. So let u=ln|z|, u is hormonic in the domain of punctured plane $\displaystyle R^2 - {0}$.
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.

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