Recipe

Classify an isolated singularity

Use when: you have an isolated singularity of f at z_0 and need to know whether it is removable, a pole, or essential

Goal: a firm classification of z_0, plus, when it's a pole, its exact finite order m

Taught in Isolated singularities -- removable, pole, and essential

1. 1

   Check that f is analytic on some punctured disk 0<|z-z_0|<r around z_0. No other bad points may crowd arbitrarily close to z_0.

   Example

   For f(z)=\dfrac{1-\cos z}{z^4} at z_0=0: f is analytic everywhere except at z=0, since \cos z is entire and only z^4 vanishes there. A punctured disk of any radius around 0 works.

   Why: without an isolated punctured disk, a Laurent series does not even exist at z_0 to classify.

2. 2

   Try to simplify f, or bound |f(z)| as z\to z_0. If it settles to a finite value, the singularity is removable and you are done.

   Example

   Here 1-\cos z\sim\dfrac{z^2}{2} near 0, while the denominator is z^4, a higher power. The ratio behaves like \dfrac{1}{2z^2}. That blows up rather than settling.

   • f simplifies or stays bounded near z_0 — removable. Assign f(z_0) that limiting value and stop here.
   • |f(z)|\to\infty as z\to z_0 instead — continue to Step 3.

   Why: a removable singularity is exactly the case where the Laurent series has no negative powers. Boundedness is the fastest test for that, with no need to build the series.

3. 3

   Test whether \phi(z)=(z-z_0)^mf(z) becomes analytic and nonzero at z_0. Start at m=1 and raise m by one each time.

   Example

   Testing m=1: zf(z)=\dfrac{1-\cos z}{z^3}\sim\dfrac{1}{2z} near 0. It is still unbounded, so m=1 fails. Testing m=2: z^2f(z)=\dfrac{1-\cos z}{z^2}\to\dfrac12 as z\to0. That is finite and nonzero.

   • some finite m makes \phi analytic and nonzero at z_0 — pole of order exactly that m.
   • no finite m ever works, and the search never ends — essential singularity.

   Why: a pole of order m is defined by this property. It is the smallest power that clears every negative term in one multiplication.

4. 4

   State whether z_0 is removable, a pole of order m, or essential. Each answer points to a different next tool.

   Why: only a true finite-order pole can use the derivative shortcut formula from the next section. An essential singularity always needs the full Laurent series.