A harmonic function on an annulus

Question

I have been working on the following problem for quite seem to figure out the end. So far, I have proven that for a real valued harmonic function $u$ on the annulus $A=\big\{z:\rho<|z|<R\big\}$, there exists bi-infinite sequences of real numbers $\big\{a_n,b_n\big\}$ so that if $z = re^{i\theta}$, then we have a locally uniformly convergent expansion: $$u(z) = a_0+b_0\log r +\sum_{n=1}^\infty [(a_nr^n+a_{-n}r^{-n})\cos n \theta +(b_nr^n +b_{-n}r^{-n})\sin n\theta ]$$ The last part of the exercise asks to give a condition, in terms of this expansion, for $u$ to have a single valued harmonic conjugate in $A$. Is there something trivial that I am missing here? Any hints or suggestions would be greatly appreciated.