Singularities
There are exactly three ways for a function to be undefined at an isolated point, and the Laurent series tells them apart at a glance. The third, the essential one, is the reason a pure delay cannot be written with poles and zeros.
01Three ways of being undefined
A point \( z_0 \) is an isolated singularity of \( f \) if the function is analytic on a neighborhood of \( z_0 \) except at \( z_0 \) itself. The classification is made by looking at what happens as you approach, and there are exactly three possible cases.
| Type | What \( |f| \) does on approach | Example |
|---|---|---|
| Removable | Tends to a finite value | \( \dfrac{\operatorname{sin} z}{z} \) at \( z = 0 \) |
| Pole of order \( m \) | Tends to infinity like \( 1/|z-z_0|^m \) | \( \dfrac{1}{z^2} \) at \( z = 0 \) |
| Essential | Tends to nothing: it depends on the direction of approach | \( e^{1/z} \) at \( z = 0 \) |
The removable singularity is so called because it fixes itself: it is enough to define \( f(0) = 1 \) for \( \operatorname{sin}(z)/z \) to be analytic on the whole plane. The pole is the one that appears in every transfer function. The essential one is the odd one, and it is the reason a pure delay resists any rational treatment.
02The series that decides it: Laurent
Around an isolated singularity, the Taylor series is not enough —it does not exist— but there is a expansion with negative powers, valid in an annulus \( 0 \lt |z - z_0| \lt \rho \):
The whole classification is in the principal part, and it can be read at a glance:
| Principal part | Type of singularity |
|---|---|
| Has no terms | Removable |
| Has \( m \) terms, the last one \( b_m \ne 0 \) | Pole of order \( m \) |
| Has infinitely many terms | Essential |
The case of \( e^{1/z} \) is self-evident. Substituting in the exponential series:
And the coefficient \( b_1 \) —the one that goes with \( 1/(z-z_0) \)— has a name of its own: it is the residue, the only term that survives integration over a closed contour. The next topic deals with it.
03Recognizing the type by looking
There is an experimental way to classify a singularity without expanding any series: surround it with a small circle and look at what it turns into. If the image shrinks, it is removable. If it grows like a power and winds \( m \) times around the origin, it is a pole of order \( m \). If it scatters over the whole plane with no pattern, it is essential.
On the left, a circle of radius \( r \) around the origin; on the right, its image, with the modulus logarithmically compressed so that everything fits. The order is estimated by comparing the size at two different radii.
04How to determine the order without drawing
For a rational function it is enough to factor the denominator and count multiplicities. For the rest there is a direct criterion: \( z_0 \) is a pole of order \( m \) if the limit
The other route is to write \( f = g/h \) with \( g \) and \( h \) analytic: if \( h \) has a zero of order \( m \) at \( z_0 \) and \( g(z_0) \ne 0 \), then \( f \) has a pole of order \( m \). And if \( g \) also vanishes there, the orders cancel: that is why \( \operatorname{sin}(z)/z \) has no pole, because the sine is also zero at the origin.
\( \sqrt{z} \) and \( \ln z \) are not defined at the origin, but their singularity is not isolated: they carry a cut. Going once completely around zero, the function does not return to the same value, so there is no Laurent series valid on an entire annulus. They are called branch points and are treated separately, by choosing a branch. In the lab it is clear: the image of the circle does not close.
05Where each one appears in electronics
| Situation | Singularity | Consequence |
|---|---|---|
| Pole of \( H(s) \) | Simple pole | A term \( e^{p t} \) in the response |
| Double pole, as in a critically damped system | Pole of order 2 | A \( t\,e^{pt} \) appears: the response is delayed |
| Pole-zero cancellation | Removable | The mode exists but is not seen from the output |
| Pure delay \( e^{-sT} \) | Essential at infinity | Not rational: it cannot be written with poles and zeros |
| \( \sqrt{s} \) from a skin effect | Branching | Models with fractional order |
The delay row explains something often run into in control: an \( e^{-sT} \) cannot be put in a pole-zero diagram, and that is why it is approximated with a rational function —the Padé approximation— in order to design. The price of that approximation is that it holds only over a band of frequencies.
The cancellation row is also worth a look. If a pole and a zero coincide exactly, the singularity is removable and disappears from the transfer function. But the physical mode is still in the circuit: it has simply stopped being observable from that output. It is the difference between what the oscilloscope measures and what goes on inside.
06In the lab
For \( \dfrac{z}{(z-1)^2(z+3)} \), \( \dfrac{1 - \cos z}{z^2} \) and \( z\,e^{1/z} \), decide what type of singularity there is at each point and of what order. Verify with the limit criterion from section 04.
Build an RLC tuned exactly to critical damping and compare the step response with that of a slightly overdamped circuit. Identify the \( t\,e^{pt} \) term contributed by the double pole.
Compare the Bode plot of \( e^{-sT} \) with that of its first-order Padé approximation, \( \dfrac{1 - sT/2}{1 + sT/2} \). Determine up to what frequency the phase error stays below 5°.
07Common mistakes
- Calling any point where the formula breaks down a pole. It may be removable, and then there is no pole.
- Counting the order by looking only at the denominator. If the numerator also vanishes there, the orders subtract.
- Treating \( \sqrt{z} \) or \( \ln z \) as isolated singularities. They are not: they have a cut.
- Believing that an essential singularity is “a pole of infinite order.” They are different things: near an essential singularity the function does not tend to infinity, it takes all values.
- Trusting an exact pole-zero cancellation. Real components have tolerance and the cancellation is never perfect.
08Self-assessment
What type of singularity does \( \operatorname{sin}(z)/z \) have at the origin?
Removable: the limit is 1. Defining \( f(0) = 1 \) makes it analytic on the whole plane.
How is the type read from the Laurent series?
By the number of terms in the principal part: none is removable, \( m \) is a pole of order \( m \), infinitely many is essential.
Why does \( e^{1/z} \) have an essential singularity?
Because its expansion has infinitely many negative powers. Approaching along the positive real axis it tends to infinity and along the negative one, to zero.
What is the order of the pole of \( \dfrac{z+1}{z^3} \)?
Order 3: the numerator does not vanish at the origin, so it cancels nothing.
Why can’t a pure delay be drawn with poles and zeros?
Because \( e^{-sT} \) is not rational: its singularity at infinity is essential. That is why it is approximated with Padé.
What is a branch point?
A point where the function does not return to the same value after going once around it, like the origin for \( \sqrt{z} \). It is not an isolated singularity.
09Further reading
- Ruel V. Churchill and James W. Brown. Complex Variables and Applications. 7th ed. (Spanish edition, “Variable compleja y aplicaciones”), McGraw-Hill, 2004. Laurent series and the classification of singularities, with the Casorati-Weierstrass theorem.
- Murray R. Spiegel. Complex Variables. Schaum’s Outline Series (Spanish edition, “Variable compleja”), McGraw-Hill, 2011. Worked Laurent expansions, which is where you learn to choose the right annulus.
- Katsuhiko Ogata. Modern Control Engineering. 5th ed. (Spanish edition, “Ingeniería de control moderna”), Pearson, 2010. The treatment of the pure delay and the Padé approximation, with its limits of validity.