What are the differences among the functions \(\frac{ \exp(z) - 1
}{ z }\text{,}\)\(\frac{ 1 }{ z^4 }\text{,}\) and \(\exp ( \frac 1 z )\) at \(z=0\text{?}\) None of them are defined at \(0\text{,}\) but each singularity is of a different nature. We will frequently consider functions in this chapter that are holomorphic in a disk except at its center (usually because that’s where a singularity lies), and it will be handy to define the punctured disk with center \(z_0\) and radius \(R\text{,}\)
We extend this definition naturally with \(\Dp[z_0, \infty] := \C
\setminus \{ z_0 \}\text{.}\) For complex functions there are three types of singularities, which are classified as follows.
If \(f\) is holomorphic in the punctured disk \(\Dp[z_0, R]\) for some \(R>0\) but not at \(z=z_0\text{,}\) then \(z_0\) is an isolated singularity of \(f\text{.}\) The singularity \(z_0\) is called
removable if there exists a function \(g\) holomorphic in \(D[z_0,R]\) such that \(f=g\) in \(\Dp[z_0, R]\text{,}\)
which is entire (because this power series converges in \(\C\)), agrees with \(f\) in \(\C \setminus \{ 0 \}\text{.}\) Thus \(f\) has a removable singularity at 0.
In Example 8.3.4, we showed that \(f: \C \setminus \{ j \pi : \, j \in \Z \}
\to \C\) given by \(f(z) = \frac 1 { \sin(z) } - \frac 1 z\) has a removable singularity at 0, because we proved that \(g: D[0,\pi] \to
\C\) defined by
\begin{equation*}
g(z) = \begin{cases}\frac 1 { \sin(z) } - \frac 1 z \amp
\text{ if } z \ne 0 \, , \\ 0
\amp \text{ if } z=0 \end{cases}
\end{equation*}
is holomorphic in \(D[0,\pi]\) and agrees with \(f\) on \(\Dp[0,\pi]\text{.}\)
\(z_0\) is a pole if and only if it is not removable and \(\lim_{z \to z_0} \left( z - z_0 \right)^{n+1} f(z) =
0\) for some positive integer \(n\text{.}\)
Suppose that \(z_0\) is a removable singularity of \(f\text{,}\) so there exists a holomorphic function \(h\) on \(D[z_0,R]\) such that \(f(z) = h(z)\) for all \(z \in \Dp[z_0,R]\text{.}\) But then \(h\) is continuous at \(z_0\text{,}\) and so
Conversely, suppose that \(\lim_{z \to z_0} \left( z -
z_0 \right) f(z) = 0\) and \(f\) is holomorphic in \(\Dp[z_0,R]\text{.}\) We define the function \(g: D[z_0,R] \to \C\) by
Suppose that \(z_0\) is a pole of \(f\text{.}\) Since \(f(z)\to\infty\) as \(z\to z_0\) we may assume that \(R\) is small enough that \(f(z)\ne 0\) for \(z\in \Dp[z_0,R]\text{.}\) Then \(\frac1f\) is holomorphic in \(\Dp[z_0,R]\) and
is holomorphic. By Theorem 8.2.1, there exist a positive integer \(n\) and a holomorphic function \(h\) on \(D[z_0,R]\) such that \(h(z_0) \ne 0\) and \(g(z) = (z-z_0)^n \, h(z) \text{.}\) Actually, \(h(z) \ne 0\) for all \(z \in D[z_0,R]\) since \(g(z) \ne 0\) for all \(z \in \Dp[z_0,R]\text{.}\) Thus
Conversely, suppose \(z_0\) is not a removable singularity and \(\lim_{z \to z_0} (z-z_0)^{n+1} f(z) = 0\) for some non-negative integer \(n\text{.}\) We choose the smallest such \(n\text{.}\) By part a), \(h(z) := (z-z_0)^n f(z)\) has a removable singularity at \(z_0\text{,}\) so there is a holomorphic function \(g\) on \(D[z_0,R]\) that agrees with \(h\) on \(\Dp[z_0,R]\text{.}\) Now if \(n=0\) this just says that \(f\) has a removable singularity at \(z_0\text{,}\) which we have excluded. Hence \(n>0\text{.}\) Since \(n\) was chosen as small as possible and \(n-1\) is a non-negative integer less than \(n\text{,}\) we must have \(g(z_0) = \lim_{z \to z_0} (z-z_0)^n f(z)
\ne 0\text{.}\) Summarizing, \(g\) is holomorphic on \(D[z_0,R]\) and non-zero at \(z_0\text{,}\)\(n>0\text{,}\) and
\begin{equation*}
f(z) = \frac{g(z)}{(z-z_0)^n} \quad \text{ for all } \quad
z\in\Dp[z_0,R]\, \text{.}
\end{equation*}
Suppose \(f\) is holomorphic in \(\Dp[z_0,R]\text{.}\) Then \(f\) has a pole at \(z_0\) if and only if there exist a positive integer \(m\) and a holomorphic function \(g: D[z_0,R] \to \C\text{,}\) such that \(g(z_0) \ne 0\) and
\begin{equation*}
f(z) \ = \ \frac{ g(z) }{ (z-z_0)^m } \qquad \text{ for all } \quad
z \in \Dp[z_0,R] \, \text{.}
\end{equation*}
The only part not covered in the proof of Proposition 9.1.6 is uniqueness of \(m\text{.}\) Suppose \(f(z)=(z-z_0)^{-m_1}g_1(z)\) and \(f(z)=(z-z_0)^{-m_2}g_2(z)\) both work, with \(m_2>m_1\text{.}\) Then \(g_2(z)=(z-z_0)^{m_2-m_1}g_1(z)\text{,}\) and plugging in \(z=z_0\) yields \(g_2(z_0)=0\text{,}\) violating \(g_2(z_0)\ne0\text{.}\)
This definition, naturally coming out of Corollary 9.1.7, parallels that of the multiplicity of a zero, which naturally came out of Theorem 8.2.1. The two results also show that \(f\) has a zero at \(z_0\) of multiplicity \(m\) if and only if \(\frac 1 f\) has a pole of order \(m\text{.}\) We will make use of the notions of zeros and poles quite extensively in this chapter.
You might have noticed that the Proposition 9.1.6 did not include any result on essential singularities. Not only does the next theorem make up for this but it also nicely illustrates the strangeness of essential singularities. To appreciate the following result, we suggest meditating about its statement over a good cup of coffee.
If \(z_0\) is an essential singularity of \(f\) and \(r\) is any positive real number, then every \(w \in
\C\) is arbitrarily close to a point in \(f(\Dp[z_0,r])\text{.}\) That is, for any \(w \in \C\) and any \(\epsilon > 0\) there exists \(z \in \Dp[z_0,r]\) such that \(|w-f(z)| \lt \epsilon\text{.}\)
There is a stronger theorem, beyond the scope of this book, which implies the Casorati–Weierstraß Theorem 9.1.9 It is due to Charles Emile Picard (1856–1941) and says that the image of any punctured disk centered at an essential singularity misses at most one point of \(\C\text{.}\) (It is worth coming up with examples of functions that do not miss any point in \(\C\) and functions that miss exactly one point. Try it!)
Invoking Proposition 9.1.6 again, we conclude that the function \(f(z) - w\) has a pole or removable singularity at \(z_0\text{,}\) which implies the same holds for \(f(z)\text{,}\) a contradiction.
\(z_0\) is a pole if and only if there are finitely many negative exponents, and the order of the pole is the largest \(k\) such that \(c_{-k} \ne 0\text{;}\)
Suppose \(z_0\) is removable. Then there exists a holomorphic function \(g: D[z_0,R] \to \C\) that agrees with \(f\) on \(\Dp[z_0,R]\text{,}\) for some \(R > 0\text{.}\) By Theorem 8.1.8, \(g\) has a power series expansion centered at \(z_0\text{,}\) which coincides with the Laurent series of \(f\) at \(z_0\text{,}\) by Corollary 8.3.7.
Conversely, if the Laurent series of \(f\) at \(z_0\) has only nonnegative powers, we can use it to define a function that is holomorphic at \(z_0\text{.}\)
Suppose \(z_0\) is a pole of order \(n\text{.}\) Then, by Corollary 9.1.7, \(f(z)=(z-z_0)^{-n}g(z)\) on some punctured disk \(\Dp[z_0,R]\text{,}\) where \(g\) is holomorphic on \(D[z_0,R]\) and \(g(z_0)\ne0\text{.}\) Thus \(g(z)=\sum_{k\ge0}c_k(z-z_0)^k\) in \(D[z_0,R]\) with \(c_0\ne0\text{,}\) so
where \(c_{-n} \not= 0\text{.}\) Define \(g(z) \ := \ \s c_{k-n} (z-z_0)^k\text{.}\) Then \(g\) is holomorphic at \(z_0\) and \(g(z_0) =
c_{-n}\ne0\) so, by Corollary 9.1.7, \(f\) has a pole of order \(n\) at \(z_0\text{.}\)
