3.3 Isolated Singularities
Suppose \(U\subseteq \mathbb {C}\) is an open subset and \(z_0\in U\). If \(f\in A(U\setminus \{ z_0\} )\), then we say that \(z_0\) is an isolated singularity of \(f\).
Often we don’t explicitly give the open set \(U\), just the function \(f\). So, in other words, we say that \(f\) has an isolated singularity at \(z_0\) if the domain of analyticity of \(f\) contains a punctured neighborhood \(D(z_0, r)\) of \(z_0\) (with \(r {\gt} 0\)).
Take \(U = \mathbb {C}\) and \(z_0 = 0\). Then the function \(f: \mathbb {C}\setminus \{ 0\} \to \mathbb {C}\) given by \(f(z) = 0\) has an isolated singularity at \(0\).
The function \(f(z) = 1/z\) also has an isolated singularity at \(0\). Here, again, \(U = \mathbb {C}\) and \(z_0 = 0\).
The function \(f(z) = e^{1/z}\) also has an isolated singularity at \(0\). Yet again, \(U = \mathbb {C}\) and \(z_0 = 0\).
The function \(\displaystyle f(z) = \frac{z+1}{z^2(z^2 + 1)}\) has isolated singularities at \(z = 0\) and \(z = \pm i\).
The function \(\operatorname{\mathrm{Log}}z\) does not have an isolated singularity at \(0\). The point is that \(\operatorname{\mathrm{Log}}z\) has domain \(\mathbb {C}\setminus \{ \text{nonnegative reals}\} \), which does not contain any punctured neighborhood of \(0\).
We say that \(z_0\) is a removable singularity of \(f\) if there exists an \(r {\gt} 0\) and an analytic function \(g\in A(D(z_0, r))\) such that \(g(z) = f(z)\) for all \(z\in D^*(z_0,r)\).
In Example 3.16 (a), the function \(f(z) = 0\) obviously has a removable singularity at \(0\). Less obviously, the function \(\displaystyle f(z) = \frac{\sin z}{z}\) has a removable singularity at \(z = 0\). To see this, note that \(\operatorname{\mathrm{ord}}_{0} \sin z = 1\). So, \(\sin z = zg(z)\) for some entire function \(g\). It follows that \(g(z) = (\sin z)/z\) for \(z\neq 0\). So \(0\) is a removable singularity of \((\sin z)/z\).
We can also talk about isolated singularities and removable singularities at \(\infty \).
We say that \(f\) has an isolated singularity at \(\infty \) if the domain of definition of \(f\) contains \(\{ z: |z| {\gt} R\} \) for some real number \(R\).
For example, any entire function \(f\) has an isolated singularity at \(\infty \). Note that, \(f\) has an isolated singularity at \(\infty \) if and only if the function \(g(z) = f(1/z)\) has an isolated singularity at \(0\).
We say that \(f\) has a removable singularity at \(\infty \) if the function \(g(z) = f(1/z)\) has a removable singularity at \(0\).
Note that, if \(f\) has an isolated singularity at \(\infty \). Then, using the notation of Definition 3.20 above, \(\lim _{z\to \infty } f(z) = g(0)\).
The function \(f(z) = 1/z\) has a removable singularity at \(\infty \). More interestingly, so does the function \(\displaystyle f(z) = \frac{z+1}{z^2(z^2 + 1)}\) from Examples 3.16 (d). In this case, we have
which has a removable singularity at \(0\).