- Let h be a harmonic function on the punctured disk
\[ U : = \left\{ z \in \mathbb { C } : 0 < | z | < 1 \right\} . \]
Show that there exists a constant \(c \in \mathbb { R }\) and a holomorphic function f on U such that \(\mathrm { R e } f ( z ) = h ( z ) + c \log | z |\) for all \(z \in U\)
Solution
Let \(L=\{w\in\mathbb C:\operatorname{Re}w<0\}\). The exponential maps \(L\) onto \(U\), and two points of \(L\) have the same exponential exactly when their difference is \(2\pi i k\) for an integer \(k\).
1. The lifted harmonic function is the real part of a holomorphic function on \(L\).
Define \(\widetilde h(w)=h(e^w)\). Composition with the holomorphic map \(w\mapsto e^w\) preserves harmonicity: the chain rule gives \[ \Delta\widetilde h(w)=|e^w|^2(\Delta h)(e^w)=0. \] On the simply connected half-plane \(L\), the harmonic function \(\widetilde h\) has a holomorphic conjugate, so there is a holomorphic \(H\) with \(\operatorname{Re}H=\widetilde h\) [@SS03]. Explicitly, the Cauchy–Riemann equations show that \(\widetilde h_s-i\widetilde h_t\) is holomorphic for \(w=s+it\). Its primitive on \(L\), after addition of a real constant, has real part \(\widetilde h\).
2. A real linear term removes the period of this conjugate.
The function \(A(w)=H(w+2\pi i)-H(w)\) is holomorphic on \(L\). Since \(e^{w+2\pi i}=e^w\), it has real part zero. The Cauchy–Riemann equations then force its imaginary part to have both partial derivatives zero. Connectedness of \(L\) implies \(A(w)=i\beta\) for a fixed \(\beta\in\mathbb R\). Set \(c=-\beta/(2\pi)\) and \(G(w)=H(w)+cw\). Then \[ G(w+2\pi i)-G(w)=i\beta+2\pi ic=0. \] Iteration in both directions makes \(G\) invariant under every translation by \(2\pi ik\), \(k\in\mathbb Z\).
3. The periodic holomorphic function descends to the desired function on \(U\).
For \(z\in U\), choose any \(w\in L\) with \(e^w=z\) and define \(f(z)=G(w)\). Any two choices differ by \(2\pi ik\), so step 2 makes this definition independent of the choice. Near each \(z\ne0\) there is a holomorphic logarithm branch \(\ell\), obtained for example from the local inverse function theorem applied to the exponential [@SS03]. Its real part is \(\log|z|<0\), so it takes its values in \(L\). Locally \(f=G\circ\ell\), which proves holomorphy of \(f\) throughout \(U\). For any such \(w\), \[ \operatorname{Re}f(z) =\operatorname{Re}H(w)+c\operatorname{Re}w =h(z)+c\log|z|. \] The constant \(c\) is real by construction, which establishes the required representation.