Isolated Singularities of an Analytic Function

Definition: A deleted neighborhood of $z_0$ is a neighborhood around $z_0$ excluding $z_0$, that is,

\[\lbrace z: 0 \lt \vert z - z_0 \vert \lt \epsilon \rbrace\]

for some $\epsilon \gt 0$.

Definition: $f$ has an isolated singularity at $z_0$ if $f$ is analytic in a deleted neigborhood of $z_0$ but is not analytic at $z_0$.

Definition: Let $z_0$ be an isolated singularity in $f$.

• If there exists a function $g$ analytic at $z_0$ such that $f(z) = g(z)$ in a deleted neighborhood around $z_0$, then $z_0$ is a removable singularity.
• If for $z \neq z_0$, $f = A(z)/B(z)$ where $A$ and $B$ are analytic at $z_0$ and $A(z_0) \neq 0$ and $B(z_0) = 0$, we say $f$ has a pole at $z_0$.
• If $f$ has neither or pole nor a removable singularity at $z_0$, then it is an essential singularity.

Theorem (Riemann’s Principle of Removable Singularities): If $z_0$ is an isolated singularity of $f$, $\lim_{z \to z_0} (z - z_0) f(z) = 0$, then it is a removable singularity of $f$.

Corollary: If $f$ is bounded in a deleted neighborhood of an isolated singularity $z_0$, then it is a removable singularity.

Theorem: If $f$ is analytic in a deleted neighborhood of $z_0$ and there exists a positive integer $k$ such that

\[\lim_{z \to z_0} (z - z_0)^{k - 1} f(z) \neq 0\]

but

\[\lim_{z \to z_0} (z - z_0)^k f(z) = 0\]

then $f$ has a pole of order $k$ at $z_0$.

Theorem (Casorati-Weierstrass): If $f$ has an essential singularity at $z_0$, then the image of $f$ in a deleted neighborhood around $z_0$ is dense in the complex plane.

Laurent Expansions