Let \(f\) be analytic in \(\Omega: 0<|z-a|<r\) except at a sequence of poles \(a_n \in \Omega\) with \(\lim_{n \rightarrow \infty} a_n = a\). Show that for any \(w \in \mathbb C\), there exists a sequence \(z_n \in \Omega\) such that \(\lim_{n \rightarrow \infty} f(z_n) = w\).
Explain the similarity and difference between the above assertion and the Weierstrass-Casorati theorem.
Solution
- Fix \(w\in\mathbb C\). For each pole \(a_n\), the function \[ h_n(z)=\frac{1}{f(z)-w} \] is holomorphic in a punctured neighborhood of \(a_n\) and has a removable singularity there with value \(0\). Since \(h_n\) is nonconstant, the open mapping theorem implies that every sufficiently small neighborhood of \(a_n\) contains points where \(h_n\) takes arbitrarily small nonzero values. Equivalently, in every sufficiently small neighborhood of \(a_n\), the values of \(f\) come arbitrarily close to \(w\).
Choose \(z_n\) so close to \(a_n\) that \[ |z_n-a_n|<\frac1n, \qquad |f(z_n)-w|<\frac1n. \] Since \(a_n\to a\), we also have \(z_n\to a\), and by construction \(f(z_n)\to w\).
- The conclusion resembles Casorati–Weierstrass: near the distinguished point \(a\), the image is dense in \(\mathbb C\). The difference is structural. Casorati–Weierstrass concerns a single isolated essential singularity at \(a\). Here \(a\) need not even be an isolated singularity of \(f\); instead, \(a\) is an accumulation point of poles, and the density conclusion is forced by the local behavior near those poles together with their accumulation at \(a\).