oab729
Apr16-09, 11:29 PM
1. The problem statement, all variables and given/known data
Without directly verifying via the laplace equations, explain why Log|z| is harmonic in the punctured complex plane.
2. Relevant equations
3. The attempt at a solution
I thought it was because Log z is analytic on the complex plane except for the nonpositive real axis, so Log |z| would be analytic and hence harmonic since any |z| turns z into a positive real (for z=/= 0), hence Log |z| is like Log z for positive reals. But, if it's that, why would they ask to show its harmonic and not analytic.
Without directly verifying via the laplace equations, explain why Log|z| is harmonic in the punctured complex plane.
2. Relevant equations
3. The attempt at a solution
I thought it was because Log z is analytic on the complex plane except for the nonpositive real axis, so Log |z| would be analytic and hence harmonic since any |z| turns z into a positive real (for z=/= 0), hence Log |z| is like Log z for positive reals. But, if it's that, why would they ask to show its harmonic and not analytic.