Laurent series — principal part, annulus of convergence: Foundations

Before you can read a single line of the parent note, you must own about a dozen small pieces of vocabulary. This page builds every one of them from nothing, in an order where each rests on the previous. Nothing below is assumed — if the parent note used a symbol, it is defined here first.

0. The complex plane — where everything lives

Figure — Laurent series — principal part, annulus of convergence

Figure s01 — Alt text: The complex plane with horizontal real axis and vertical imaginary axis. A lavender arrow runs from the origin to the point $z = x + i y$ . A butter-yellow segment along the bottom marks the horizontal distance $x = Re (z)$ ; a mint segment up the side marks the vertical distance $y = Im (z)$ ; a coral dashed line drops from the point down to the real axis. The length of the lavender arrow is labelled $∣ z ∣ = x^{2} + y^{2}$ .

$x$ is the real part, written $Re (z)$ — the horizontal coordinate (the butter-yellow segment).
$y$ is the imaginary part, written $Im (z)$ — the vertical coordinate (the mint segment).

Why we need it: every function in this chapter eats a point $z$ of this plane and returns another complex number. There is no number line here — there is a whole plane — and that extra dimension is exactly why singularities can be "fenced off" as rings later.

Why we need it: the entire idea of an annulus is "all points whose distance from $a$ is between $r$ and $R$ ." That distance is a modulus. Whenever you see $∣ z - a ∣$ , read it aloud as "the distance from $z$ to the fixed point $a$ ."

1. The complex exponential — turning angles into points

Polar form (next section) leans on the symbol $e^{i θ}$ . Before we may use it, we must say what it means, because raising $e$ to an imaginary power is not something ordinary arithmetic explains.

Why we need it: every loop in this chapter is a circle, and $e^{i θ}$ is the cleanest way to drive a point around a circle by turning one dial $θ$ . That is why the parent's contour proof is written with $e^{i θ}$ .

2. Polar form — the secret weapon for going in circles

Figure s02 — Alt text: The complex plane with a mint dashed unit circle centred at the origin. A coral arrow of length one points at angle $θ$ (this is $e^{i θ} = cos θ + i sin θ$ , landing on the unit circle). A longer lavender arrow along the same direction reaches the point $z = ρ e^{i θ}$ , its length labelled $ρ = ∣ z ∣$ . A butter-yellow arc near the origin marks the angle $θ$ swept up from the positive real axis.

Now the tiny movement rule you will meet in the proof — with its derivation:

Reading it as a picture: nudging $θ$ a little slides $z$ along the circle; the factor $i$ turns the radial arrow $e^{i θ}$ by $9 0^{\circ}$ so the step is tangent to the circle (exactly the direction you move when going around).

3. Analytic — the word for "nice, no trouble here"

Why we need it: the Cauchy integral formula and Taylor's theorem only work where $f$ is analytic. Laurent series exist precisely to survive where analyticity fails — so you must know what "nice" means before you can talk about "not nice."

4. Singularity, pole, essential — the trouble-spots

Figure s03 — Alt text: A 3-D surface plot over the complex plane (horizontal axes $Re (z)$ and $Im (z)$ ) whose height is $∣ f (z) ∣$ for $f (z) = 1/ z$ . Away from the origin the surface is a gently sloping valley; directly above $z = 0$ it rockets upward into a tall spike labelled "blow-up", picturing the singularity.

The figure shows the height $∣ f (z) ∣$ as a surface. At a singularity it shoots up like a spike.

See Poles and singularities for the full picture. The whole point of the principal part is that it measures how bad the singularity is by counting negative-power terms.

5. Series and the summation symbol $\sum$

Why we need it: the whole innovation of Laurent over Taylor is: allow negative powers. The negative-power terms are the "principal part." The name of the game is nothing more than this split.

6. Geometric series — the engine that builds every expansion

7. Contour integral $\oint_{γ}$ , the one fact, and the residue

Figure s04 — Alt text: The complex plane showing a shaded lavender ring (annulus) around a fixed centre point $a$ marked with a slate dot. A coral inner circle has radius $r$ (a coral arrow from $a$ marks that inner radius); a lavender outer circle has radius $R$ . A mint dashed circle sits inside the ring as the loop $γ$ , with a mint arrowhead showing it is traversed anticlockwise (positive orientation).

Why we need it: this is the "sieve" that pulls out one coefficient at a time in the parent's derivation. Only the power $k = - 1$ leaves a residue behind; all others integrate to zero over a full loop.

8. The annulus — the ring these series live on

The inner radius $r$ = the distance from $a$ to the closest singularity of $f$ (the one that fences off the hole in the middle). The outer radius $R$ = the distance from $a$ to the next singularity further out. See Annulus of convergence. Between them, $f$ is analytic, and that ring is exactly where its Laurent series converges.

How the pieces feed the topic

The diagram below is a dependency map: read it top-to-bottom, where each arrow means "you need the thing at the tail before you can do the thing at the head." If your renderer does not draw the diagram, here is the same information in words: the complex plane ( $z = x + i y$ ) gives you the modulus (distance), which builds the annulus. The complex exponential $e^{i θ}$ gives you polar form, which lets you loop around $a$ ; combined with the contour fact (only $k = - 1$ survives) this yields the coefficient formula and residue. Separately, analytic defines what a singularity is, and poles/essential classify it. The geometric series is the algebra engine that produces the actual expansions. All four strands converge on the Laurent series.

Equipment checklist

Test yourself — you should be able to answer each before reading the parent note.

What does the fixed symbol

a

denote throughout this chapter?

One chosen, fixed point of the complex plane — the centre we expand around and measure all distances

∣ z - a ∣

from.

What does

∣ z - a ∣

mean in one plain phrase?

The distance in the plane from the point

z

to the fixed point

a

State Euler's formula and what

e^{i θ}

looks like geometrically.

e^{i θ} = cos θ + i sin θ

; it is the point on the unit circle reached by walking angle

θ

anticlockwise.

Write

z

in polar form and say what each symbol is.

z = ρ e^{i θ}

;

ρ = ∣ z ∣ \geq 0

is the distance from

0

, and

θ

is the angle from the positive real axis (defined only up to adding multiples of

2 π

Show that

\frac{d}{d θ} e^{i θ} = i e^{i θ}

Differentiate

cos θ + i sin θ

to get

- sin θ + i cos θ = i (cos θ + i sin θ)

What does "analytic" (holomorphic) mean precisely?

Complex-differentiable throughout a small disk around the point; equivalently, equal to its own Taylor series there.

What is a disk in the complex plane?

The filled set of all points within a fixed distance

R

of the centre,

{z : ∣ z - a ∣ < R}

Give the precise definition of "pole of order

m

The smallest

m \in N

such that

(z - a)^{m} f (z)

is analytic and nonzero at

a

State the geometric series and its convergence condition.

\frac{1}{1 - w} = \sum_{n \geq 0} w^{n}

, valid only when

∣ w ∣ < 1

Parametrise a circle and evaluate

\oint_{γ} (z - a)^{k} d z

With

z = a + ρ e^{i θ}

it becomes

i ρ^{k + 1} \int_{0}^{2 π} e^{i (k + 1) θ} d θ

, giving

2 π i

k = - 1

and

0

otherwise.

Write the full Laurent series and its coefficient formula.

f (z) = \sum_{n = - \infty}^{\infty} c_{n} (z - a)^{n}

with

c_{n} = \frac{1}{2 π i} \oint_{γ} \frac{f ( z )}{( z - a ) ^{n + 1}} d z

What is the residue

Res_{f} (a)

in terms of the Laurent coefficients?

It is

c_{- 1}

, the coefficient of the

\frac{1}{z - a}

term.

What shape is an annulus, are its boundaries included, and what do its two radii mean?

A ring/donut

r < ∣ z - a ∣ < R

with both rims excluded (strict inequalities);

r

is the distance from

a

to the nearest singularity,

R

to the next one out.

What distinguishes a Laurent series from a Taylor series?

The Laurent series is also allowed negative powers

(z - a)^{- n}