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## Homework Statement

Find harmonic conjugate of u(x,y)=ln(x^2+y^2) and specify the region it is defined

then show u has no harm conj on C\{0}

## Homework Equations

## The Attempt at a Solution

Ok so i found the harmonic conj by converting to polar and found it to be v(r,Θ) = Θ.

I am having trouble finding out where v is defined. Also it seems our analytic function f = u + iv

is log(z) since u(r,Θ) = ln|r|. Isn't that defined everywhere but zero? Why would u not have a harmonic conj at C\{0}?