CONFORMAL MAPS AND MOBIUS
TRANSFORMATIONS
By Mariya Boyko
OVERVIEW
Introduction And Basic Definitions
 Basic Topology
 Complex Analysis
 Understanding Riemann’s Theorem
 Mobius Transformations
 How To Find Mobius Transformations
 Example

INTROUCTION AND BASIC DEFINITIONS
Complex functions of a complex variable require
4 dimensions to graph.
 Instead, we use two Cartesian planes, one for the
domain, and one for the range.
 In 1851 Riemann in his PhD thesis found a
theorem now known as the Riemann Mapping
Theorem

Let D be any simply connected domain in the
complex plane which is not the whole plane itself.
Then there exists a one-to-one conformal map
which takes D onto the open unit disc |z|<1.
BASIC TOPOLOGY
A Neighbourhood of a point z is a disc centered
around z.
 Interior Point: An interior point of a set S is a
point for which there exists a neighbourhood
entirely in S.
 Boundary Point: A boundary point of a set S is a
point for which every neighbourhood has points
in S and in S’
 Open Set: A set is called open if all its points are
interior points. An equivalent definition is if it
does not contain its boundary.

BASIC TOPOLOGY
Connected Set: A set is called connected if it
cannot be expressed as the union of two disjoint,
non-empty open sets.
 Domain: A connected open set.
 Simply connected: Every loop in the set can be
continuously deformed to a point. (ie. it does not
have any holes)

COMPLEX ANALYSIS
A complex function which is differentiable is
called a holomorphic or analytic function.
 Holomorphic is a very strong condition which has
many implications.
 A function (map) from an open set U is called
conformal if it is holomorphic on U and whose
derivative is nowhere zero on U.
 Conformal maps preserve angles.
 The composition of conformal maps is conformal
and the inverse of a conformal map is again
conformal

UNDERSTANDING RIEMANN'S THEOREM
Theorem:
Let D by any simply connected domain in the
complex plane which is not the whole plane itself.
Then there exists a one-to-one conformal map
which takes D onto the open unit disc |z|<1.
Let F,G be two simply connected domains which
are not the entire complex plane. By Riemann’s
theorem, there exist two one-to-one conformal
maps f,g mapping F,G respectively onto the unit
disc |z|<1.
 Consider the map g-1∘f. It is a one-to-one
conformal map taking F to G.

UNDERSTANDING RIEMANN’S THEOREM
Theorem:
Let D by any simply connected domain in the
complex plane which is not the whole plane itself.
Then there exists a one-to-one conformal map
which takes D onto the open unit disc |z|<1.
Thus by Riemann’s Theorem there always exists
a one-to-one conformal map between any two
simply connected domains in the complex plane.
 Even though the theorem tells us that such a
map exists, it does not tell us how to find it.

UNDERSTANDING RIEMANN’S THEOREM
Theorem:
Let D by any simply connected domain in the
complex plane which is not the whole plane itself.
Then there exists a one-to-one conformal map
which takes D onto the open unit disc |z|<1.

In fact, even simple mappings such as the one
from the open unit square to the open unit disc
cannot be expressed in terms of elementary
functions.
MOBIUS TRANSFORMATIONS
The extended complex plane is the complex plane
together with a point called “infinity”
 The map defined by f(z) =(az + b)/(cz + d) from
the extended complex plane to the extended
complex plane where a,b,c and d are complex
numbers and ad ≠ bc is called a Möbius
transformation
 It is a composition of translations, rotations,
magnifications and inversions.

MOBIUS TRANSFORMATIONS
These simple one-to-one conformal functions map
circles and lines to circles and lines and therefore
the same is true of all Möbius transformations.
 The pole of a Möbius transformation is the point
where the denominator is zero.
 We define f(-d/c) = ∞ and f(∞) = a/c
 If a line or circle passes through the pole of the
Möbius transformation then it always gets
mapped to a line, otherwise it is mapped to a
circle

HOW TO FIND MOBIUS
TRANSFORMATIONS
To completely specify a Möbius transformation,
all we need is to define the image of three distinct
points.
 Let f(z) be a Möbius transformation and suppose
that z1, z2 and z3 are three distinct points in the
complex plane. Suppose also that f(z1) = w1, f(z2)
= w2 and f(z3) = w3. There are two possibilities,
either w1, w2 and w3 are non-collinear and they
determine a unique circle or else they determine
a unique line. (if wi = ∞ then the image is always
a line)

EXAMPLE
Let C1 and C2 be two circles in the complex plane.
How can we find a Möbius transformation from
one to the other?
 Pick any three distinct points on C1, say a, b and
c. We will first find the Möbius transformation
T(z) which maps C1 to the real axis. To do this,
let T(a) = 0, T(b) = 1 and T(c) = ∞. It is not hard
to verify that T(z) =[(z – a)(b – c)]/[(z - c)(b - a)] is
the required Möbius transformation.

EXAMPLE (CONTINUED)
The function [(z – a)(b – c)]/[(z - c)(b - a)] is called
the cross-ratio of z, a, b and c and is denoted by
(z,a,b,c)
 Now take three distinct points on C2, say d, e and
f and in the same way find a Möbius
transformation S(w) which maps C2 to the real
axis. We let S(d) = 0, S(e) = 1 and S(f) = ∞ and we
obtain S(w) = [(w - d)(e - f)]/[(w - f)(e - d)] . Then,
since Möbius transformations are closed under
composition and inverses (as can be directly
verified with a calculation), g(z) = S-1(T(z)) is the
required Möbius transformation.

EXAMPLE (CONTINUED)
To find g(z) explicitly, notice that g(a)= S-1(T(a))=
S-1(0) = d. In the same way, g(b) = e and g(c) = f.
Thus in general, w = g(z) = S-1(T(z)). This implies
that S(w) = T(z). Therefore we just equate the
two cross-ratios (z, a, b, c) and (w, d, e, f) and
solve for w in terms of z.
 To map a line to circle, line to line or circle to
line, we follow the same process with only a
minor modification to the Möbius
transformations S(w) and T(z).
