Download Complex numbers, The Riemann sphere From earlier courses we

Complex numbers, The Riemann sphere
From earlier courses we have two ways of representing complex numbers: rectangular coordinates and polar coordinates. In rectangular
coordinates, a complex number z is represented as z = x + iy, where x, y are real numbers. This way of viewing complex numbers is
the most common one, and it works particularly well for addition and subtraction, since this
becomes simply vector calculus. On the other
hand, multiplication of complex numbers has
no particularly natural representation in rectangular coordinates. For this purpose, polar coordinates can be more useful. In polar coordinates, a complex number z is expressed as
z = reiθ , where r = |z| and θ is the argument
of z, i.e. the angle from the positive real axis to
the vector representing z, measured in the positive direction. In polar coordinates, multiplication and division are very simple to express:
If z1 = r1eiθ1 and z2 = r2eiθ2 , then multiplication is expressed by multiplying r1 and r2,
and adding θ1 and θ2. However, it is very important in this course to note that the polar
coordinates are not uniquely determined: Adding an integer multiple of 2π to θ gives new
polar coordinates which correspond to exactly
the same point.
We will in this lecture consider a third way
of representing complex number, namely as
points on the unit sphere S in R3.
In the picture, we see how an arbitrary complex number z is mapped onto the intersection point (x1, x2, x3) between the unit sphere and the straight line passing through z and
the northpole N. This map (“the stereographic
projection”) defines a bijection between C and
S \ {N}. (Note that we have identified the real
and imaginary axis with the x1- and x2-axis.
When z → ∞, the corresponding projection on
S tends to the northpole N . For this reason, it
b
is very natural to identify S with C∪{∞} (=C).
Thus, this sphere (the Riemann sphere) can be
viewed as a compact extension of C.
On S, the ordinary algebraic operations on complex numbers look rather awkward. There are
however other situations when the Riemann
sphere is quite convenient. Let us start by computing the projection explicitly: To find the
point (x1, x2, x3), we start by constructing the
line passing through (0, 0, 1) and (x, y, 0), which
gives (x1, x2, x3) = (tx, ty, 1 − t). Then, we de2 + x2 = 1
termine t by the condition x2
+
x
1
2
3
2
which gives t = 2/(|z| + 1). Inserting in the
parametric expression for the line we get
2Re z
x1 = 2
,
|z| + 1
2Im z
x2 = 2
,
|z| + 1
|z|2 − 1
x3 = 2
|z| + 1
We can also eliminate t to get for x, y:
x1
x1
x=
,
y=
.
1 − x3
1 − x3
(1)
Theorem 1 Both lines and circles in C correb
spond to circles on C.
Note that a circle on S is given by the intersection of S with a plane. The theorem is proved
by noting that every circle AND line in C can
be written on the form
A(x2 + y 2) + Cx + Dy + E = 0
(2)
(A 6= 0 gives circles and A = 0 gives lines.)
By inserting (1) into (2) we arrive after some
work at the intersection of S with the plane
A(1 + x3) + Cx1 + Dx2 + E(1 − x3) = 0.
Definition 1 A Möbius map is a function of
the form
az + b
(where ad − bc 6= 0).
f (z) =
cz + d
A Möbius map can in a natural way be consib
dered as a bijective map from Cb → C.
Theorem 2 A Möbius map maps a line or a
circle in C onto a line or a circle.
The proof consists in splitting f (z) up into
a composition of translations and dilatations
(which obviously preserve the class of lines and
circles), and an inversion map z 7→ 1/z which
on S can be shown to correspond to a rotation
(which obviously maps circles on S onto circles
on S).