Zeros and Singularities

In this lesson you’ll classify isolated singularities into three types, learn several equivalent tests for each, determine the order of a pole quickly, and see the extraordinary behaviour near an essential singularity.

If $f$ is analytic at $z_0$ and $f(z_0)=0$, then near $z_0$

$$f(z) = (z-z_0)^m g(z), \qquad g \text{ analytic}, \quad g(z_0)\ne0$$

and $m$ is the order of the zero. Equivalently $m$ is the smallest index with $f^{(m)}(z_0)\ne0$, or the lowest power appearing in the Taylor series.

Orders behave simply: they add under multiplication and subtract under division. That single rule handles most classification problems, as we will see.

Zeros of a non-zero analytic function are isolated, which we saw last lesson.

Suppose $f$ is analytic on a punctured disc $0<|z-z_0|<R$. Exactly one of three things happens.

Removable. $\lim_{z\to z_0}f(z)$ exists and is finite. Then defining $f(z_0)$ to be that limit makes $f$ analytic at $z_0$, and the Laurent series has no principal part. The classic case is $\frac{\sin z}{z}$ at 0, whose limit is 1.

Riemann’s theorem makes this remarkably cheap to check: boundedness near $z_0$ is enough. You do not need the limit to exist a priori; if $|f|$ is merely bounded on a punctured neighbourhood, the singularity is removable and the limit exists automatically.

Pole of order $m$. $|f(z)|\to\infty$ as $z\to z_0$, and the principal part has exactly $m$ terms with lowest power $(z-z_0)^{-m}$. Equivalently $(z-z_0)^mf(z)$ has a removable singularity with nonzero limit, or

$$f(z) = \frac{g(z)}{(z-z_0)^m}, \qquad g \text{ analytic}, \quad g(z_0)\ne0$$

Essential. Neither of the above: $|f|$ has no limit, finite or infinite. The principal part has infinitely many terms. The standard example is $e^{1/z}$ at 0.

The figure separates the three by measurement. On a circle of radius $r$ about the origin, $\frac{\sin z}{z}$ keeps $|f|$ near 1, $\frac{1}{z^2}$ has $|f|$ exactly $1/r^2$ in every direction, and $e^{1/z}$ has $|f| = e^{\cos\theta/r}$, which is both enormous and minuscule on the same circle. That last behaviour is what “essential” means in practice.

For $f = \frac{p}{q}$ with $p,q$ analytic at $z_0$, let $a$ be the order of the zero of $p$ there and $b$ the order for $q$ (either may be 0). Then:

• $a \ge b$: removable
• $a < b$: pole of order $b-a$

So classification is arithmetic on orders. This handles almost every example you will meet, and it explains cases that look surprising:

$$\frac{\sin z}{z^3}: \ a=1,\ b=3 \implies \text{pole of order } 2, \text{ not } 3$$ $$\frac{1-\cos z}{z^2}: \ a=2,\ b=2 \implies \text{removable}$$

Cancel before classifying. Failing to notice the numerator’s zero is the standard mistake here.

Essential singularities are genuinely strange, and two theorems say how strange.

Casorati–Weierstrass. In any punctured neighbourhood of an essential singularity, the image of $f$ is dense in $\mathbb{C}$: it comes arbitrarily close to every complex value.

Picard’s great theorem. Stronger still: $f$ actually attains every complex value infinitely often in any punctured neighbourhood, with at most one exception.

For $e^{1/z}$ near 0, the single exception is 0 itself, since the exponential is never zero. Every other complex number is hit infinitely many times in any disc around the origin, however small. That is worth pausing on: an arbitrarily small punctured disc maps onto essentially all of $\mathbb{C}$, infinitely many times over.

To classify behaviour at $\infty$, substitute $w = 1/z$ and examine $g(w) = f(1/w)$ at $w=0$.

• Polynomial of degree $n$: pole of order $n$ at $\infty$
• $e^z$: essential singularity at $\infty$
• $1/z$: removable at $\infty$, with value 0
• $\sin z$: essential at $\infty$

A useful classification follows: a function analytic on all of $\hat{\mathbb{C}}$ is constant, and a function meromorphic on all of $\hat{\mathbb{C}}$ is rational. So rational functions are exactly the functions with no worse than poles anywhere on the sphere.

Example 1: Classify the singularity of $\frac{\sin z}{z}$ at 0.

Solution. Numerator order 1, denominator order 1, so $a = b$ and the singularity is removable.

Explicitly,

$$\frac{\sin z}{z} = \frac{z-\frac{z^3}{6}+\cdots}{z} = 1-\frac{z^2}{6}+\cdots$$

No principal part, and the limit is 1. Defining $f(0)=1$ gives an entire function. ∎

Example 2: Classify $\frac{\sin z}{z^3}$ at 0.

Solution. $a=1$, $b=3$, so a pole of order 2.

$$\frac{\sin z}{z^3} = \frac{1}{z^2}-\frac16+\frac{z^2}{120}-\cdots$$

Confirming: the lowest power is $z^{-2}$. Note the residue is 0 here, since there is no $\frac1z$ term. ∎

A pole of order 2 with zero residue is a good reminder that order and residue are independent pieces of information.

Example 3: Classify $\frac{1-\cos z}{z^2}$ at 0.

Solution. Since $1-\cos z = \frac{z^2}{2}-\frac{z^4}{24}+\cdots$, the numerator has a zero of order 2. So $a=b=2$: removable, with limit

$$\lim_{z\to0}\frac{1-\cos z}{z^2} = \frac12$$

∎

Example 4: Classify the singularities of $\frac{z}{(z-1)^2(z+3)}$.

Solution. The numerator vanishes only at 0, which is not a singularity.

At $z=1$: $a=0$, $b=2$, so a pole of order 2.

At $z=-3$: $a=0$, $b=1$, so a simple pole.

At $\infty$: substituting $w=1/z$, the function behaves like $\frac{1/w}{(1/w)^3} = w^2 \to 0$, so $\infty$ is a removable singularity with value 0. ∎

Example 5: Classify $e^{1/z}$ at 0 and demonstrate the essential behaviour.

Solution. Its Laurent series $\sum_{n\ge0}\frac{z^{-n}}{n!}$ has infinitely many negative powers, so the singularity is essential.

Approach along the positive real axis, $z = t \to 0^+$: $e^{1/t}\to\infty$.

Approach along the negative real axis, $z = -t$: $e^{-1/t}\to0$.

Approach along the imaginary axis, $z = it$: $|e^{1/(it)}| = |e^{-i/t}| = 1$ for all $t$.

Three directions, three completely different behaviours. No limit, finite or infinite, and by Picard every nonzero value is attained infinitely often in any disc about 0.

Example 6: Classify $\frac{1}{\sin z}$.

Solution. $\sin z$ has simple zeros exactly at $z = n\pi$ for $n\in\mathbb{Z}$, so $\frac{1}{\sin z}$ has simple poles at each $n\pi$.

At $\infty$: the poles $n\pi$ accumulate there, so the singularity at $\infty$ is not isolated and the classification does not apply. ∎

This is why $\frac{1}{\sin z}$ is meromorphic on $\mathbb{C}$ but not on $\hat{\mathbb{C}}$, and consequently not rational, even though it has only poles in the finite plane.

Example 7: Use Riemann’s theorem.

$f$ is analytic on $0<|z|<1$ and $|f(z)|\le 7$ throughout. What kind of singularity is at 0?

Solution. Bounded near the singularity, so by Riemann’s theorem it is removable, and $\lim_{z\to0}f(z)$ exists.

Answer: removable. ∎

Boundedness alone is enough, which is much less than you would expect to need. It also gives the quickest proof that a pole cannot be bounded and an essential singularity cannot be either.

System poles. The poles of a transfer function are the system’s natural frequencies, and their order controls the response shape: a simple pole gives exponential decay, a double pole gives a $te^{\lambda t}$ term. Reading order off the denominator is routine engineering work.

Resonance strength. The residue at a pole sets the amplitude of the corresponding resonance, and the order sets how sharply the response peaks. Both are used to fit measured frequency-response data.

Removable singularities in code. Expressions like $\frac{\sin x}{x}$ and $\frac{1-\cos x}{x^2}$ appear constantly in graphics and numerics and must be special-cased near 0 to avoid dividing by zero. Knowing the limits, 1 and $\frac12$, is what the special case uses.

Stability boundaries. A pole crossing from the left half plane to the right marks the onset of instability, so control design is largely the management of pole locations. The classification here is the mathematics behind the root-locus plot.

Renormalization in physics. Divergences in quantum field theory are poles in a complex parameter, and their order determines how the divergence is subtracted. The vocabulary of simple and higher-order poles is used exactly as in this lesson.

Essential singularities in dynamics. Points where a map has an essential singularity generate the most chaotic behaviour, and Picard’s theorem is the formal reason: an arbitrarily small neighbourhood spreads over almost all of $\mathbb{C}$.