3.6 Classification of isolated singularities
In this section, \(f\) will be function with an isolated singularity at \(z_0\). We suppose that \(f\) is analytic in the puncture disk \(D^* = D^*(z_0, r)\), and we write
for the Laurent series of \(f\) on \(D^*\).
The principal part of \(f\) is the series \(\sum _{k {\lt} 0} a_k (z-z_0)^k\).
The principal part of \(f\) converges to a function \(f_{\mathrm{pp}}\) on \(D^*\). The function \(f - f_{\mathrm{pp}}\) has a removable singularity at \(z_0\) and is equal to
Moreover, \(f\) has a removable singularity at \(z_0\) if and only if \(f_{\mathrm{pp}} = 0\).
Essentially obvious from the definitions.
Set \(P = \{ k {\gt} 0: a_{-k} \neq 0\} \), and set \(m = \sup P\). If \(P\) is finite and nonempty, we say that \(f\) has a pole of order \(m\) at \(z_0\). We say that \(f\) has a simple pole at \(z_0\) if \(m = 1\). If \(P\) is infinite, we say that \(f\) has an essential singularity at \(z_0\).
Because of partial fractions, rational functions never have essential singularities. \(\sin 1/z\) has an essential singularity at \(0\).