Let \(f\) be holomorphic and zero-free on the annulus \[ \Omega=\{z\in\mathbb C:1<|z|<2\}. \] Show that there exist an integer \(n\) and a holomorphic function \(g\) on \(\Omega\) such that \[ f(z)=z^n e^{g(z)} \] for every \(z\in\Omega\).
Solution
Fix a radius \(\rho\) with \(1<\rho<2\), and let \[ \gamma(t)=\rho e^{it},\qquad 0\le t\le2\pi. \]
1. The number\[ n\coloneqq \frac{1}{2\pi i}\int_\gamma \frac{f'(z)}{f(z)}\,dz \]is an integer.
Since \(f\) has no zeros on \(\Omega\), the curve \(f\circ\gamma\) is a closed curve in \(\CC^\times\). By the change-of-variables formula for line integrals, \[ \frac{1}{2\pi i}\int_\gamma \frac{f'(z)}{f(z)}\,dz = \frac{1}{2\pi i}\int_{f\circ\gamma}\frac{dw}{w}. \] The last quantity is the winding number of \(f\circ\gamma\) about \(0\), hence is an integer.
2. The holomorphic function\[ h(z)\coloneqq \frac{f'(z)}{f(z)}-\frac{n}{z} \]has a holomorphic primitive on \(\Omega\).
The function \(h\) is holomorphic on \(\Omega\), so it has a Laurent expansion \[ h(z)=\sum_{k=-\infty}^{\infty}a_kz^k \] there. By step 1, \[ \int_\gamma h(z)\,dz = \int_\gamma \frac{f'(z)}{f(z)}\,dz -n\int_\gamma \frac{dz}{z} =0. \] For a Laurent series, this integral is \(2\pi i a_{-1}\), so \(a_{-1}=0\). Therefore the termwise antiderivative \[ g_0(z) = \sum_{k\ne-1}\frac{a_k}{k+1}z^{k+1} \] converges locally uniformly on the same annulus and satisfies \(g_0'=h\).
3. The function\[ q(z)\coloneqq f(z)z^{-n}e^{-g_0(z)} \]is a nonzero constant on \(\Omega\).
The function \(q\) is holomorphic and nowhere zero on \(\Omega\). Using step 2, \[ \frac{q'(z)}{q(z)} = \frac{f'(z)}{f(z)}-\frac{n}{z}-g_0'(z) =0. \] Hence \(q'=0\). Since \(\Omega\) is connected, \(q\) is constant, say \(q\equiv c\) with \(c\ne0\).
4. There is a holomorphic function \(g\) on \(\Omega\) such that\[ \boxed{f(z)=z^ne^{g(z)}} \]for every \(z\in\Omega\).
Choose \(\lambda\in\CC\) with \(e^\lambda=c\) and set \[ g\coloneqq g_0+\lambda. \] By step 3, \[ f(z)=cz^ne^{g_0(z)}=z^ne^{g(z)}. \]
5. Q.E.D.
Step 1 gives the required integer \(n\), and step 4 gives the required holomorphic function \(g\) and the factorization.