I have a problem in Ransford's book at exercise 2 in chapter 1.1. This exercise states the following
Let $h$ be a harmonic function on annulus $\{z\in\mathbb{C}\colon \rho_1<|z|<\rho_2\}$. Using the fact that $h_x-ih_y$ is analytic, prove that there exists unique constants $\{a_n\}$, $n\in\mathbb{Z}$ and $b$, with $a_0, b\in\mathbb{R}$ such that $$h(z)=\operatorname{Re}\left(\sum_{n\in\mathbb{Z}}a_nz^n\right)+b\log|z|\quad(\rho_1<|z|<\rho_2)$$
This exercise is an application of a general theorem called the Logarithmic Conjugation Theorem. One good referrence for that is in Axler's paper, where he states the general version for a general multiply connected domain, which implies the one with the annulus as in the hypothesis.
My question follows now: The theorem and how this works is understood, even the fact that $b$ is real is clear to me. A cannot find any good reason why $a_0$ is indeed real. I am trying to compute it from the Laurent expansion series formula, where it states that $$a_0=\frac{1}{2\pi i}\int_{C(0,r)}\frac{(h_x-ih_y)(z)}{z}dz\qquad (\rho_1<r<\rho_2).$$
Even if i break the complex contour integral into both imaginary and real part, the differential that appears is not exact, as in the case of $b$, however I might have done something wrong with the computation. Can you please contribute your thoughts on this little detail? I would be so much grateful.
Sincerely