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Take the punctured complex planeandIn polar coordinates, $u(r,\theta )=\log r$, soThus $u$ is harmonic on $D$.

If $u=\mathrm{Re}f$ for a holomorphic function $f$, the calculation from part (a) would giveThe integral of a derivative around a closed curve is zero, whereas the residue theorem givesThis contradiction proves that no such $f$ exists. It is precisely the log modulus has no global harmonic conjugate on the punctured plane obstruction.