48 I The Expanding Universe of Numbers
It remains to show thatπhas its usual geometric significance. Since the continu-
ously differentiable functionz(t)=eitdescribes the unit circle astincreases from 0
to 2π, the length of the unit circle is
L=
∫ 2 π0|z′(t)|dt.But|z′(t)|=1, sincez′(t)=ieit, and henceL= 2 π.
In a course of complex analysis one would now define complex line integrals,
prove Cauchy’s theorem and deduce its numerous consequences. The miracle is that,
ifD ={z ∈C:|z|<ρ}is a disc with centre the origin, then any differentiable
functionf:D→Ccan be represented by apower series,
f(z)=c 0 +c 1 z+c 2 z^2 +···,which is convergent for|z|<ρ. It follows that, if fvanishes at a sequence of dis-
tinct points converging to 0, then it vanishes everywhere. This is the basis foranalytic
continuation.
A complex-valued function fis said to beholomorphicata ∈Cif, in some
neighbourhood ofa, it can be represented as the sum of a convergent power series (its
‘Taylor’ series):
f(z)=c 0 +c 1 (z−a)+c 2 (z−a)^2 +···.It is said to bemeromorphicata∈Cif, for some integern, it can be represented near
aas the sum of a convergent series (its ‘Laurent’ series):
f(z)=c 0 (z−a)−n+c 1 (z−a)−n+^1 +c 2 (z−a)−n+^2 +···.Ifc 0 =0, then(z−a)f′(z)/f(z)→−nasz→a.Ifalson>0 we say thatais a
poleoffoforder nwithresidue cn− 1 .Ifn= 1 ,the residue isc 0 and the pole is said
to besimple.
LetGbe a nonempty connected open subset ofC. From what has been said, if
f:G→Cis differentiable throughoutG, then it is also holomorphic throughoutG.
Iff 1 and f 2 are holomorphic throughoutGand f 2 is not identically zero, then the
quotientf= f 1 /f 2 is meromorphic throughoutG. Conversely, it may be shown that
iffis meromorphic throughoutG,thenf =f 1 /f 2 for some functionsf 1 ,f 2 which
are holomorphic throughoutG.
The behaviour of many functions is best understood by studying them in the
complex domain, as the exponential and trigonometric functions already illustrate.
Complex numbers, when they first appeared, were called ‘impossible’ numbers. They
are now indispensable.
6 QuaternionsandOctonions
Quaternions were invented by Hamilton (1843)in order to be able to ‘multiply’ points
of 3-dimensional space, in the same way that complex numbers enable one to multiply