A Harmonic Function, but not the real part of a pure Function

Take the Harmonic Function $ \ log | z | $ on $ B () \ setminus \ {0 \} $, so it cannot be $ B) \ setminus \ {0 \} $ indicates the actual part of a pure function. otherwise

$ F (z) = \ log | z | + IU (z) \ In H (B (0, 1) \ setminus \ {0 \}) $

$ U (z) $ is a real value. then $ e ^ {f (z) }=| z | E ^ {IU (z) }$, so $ \ left | \ frac {e ^ {f (z) }}{ z} \ right | \ equiv1 $

Therefore, $ \ frac {e ^ {f (z) }}{ z} $ is a constant value, so $ \ Theta \ In (-\ Pi, \ Pi] $ make \ begin {Align *} e ^ {f (z)} & = ze ^ {I \ Theta} \ rightarrow U (z) & =\ Theta + {\ RM Arg} Z + 2 k (z) \ pi \ end {Align *}

$ K (z) $ is a function with a value of $ \ mathbb Z $, while $ U (z) $ indicates the continuity of $ K (z) $, then $ {\ RM Arg} z = U (Z)-\ theta-2k \ pi \ in C (B (0, 1) \ setminus \ {0 \}) $

This is not possible, because $ {\ RM Arg} $ is interrupted on the negative real axis.

A Harmonic Function, but not the real part of a pure Function