Linear Fractional Transformations

students, in complex analysis, some functions can bend and stretch the plane in very organized ways ✨. One of the most important families of such functions is the linear fractional transformations, also called Möbius transformations. These transformations are central to conformal mapping because, where they are defined, they preserve angles and turn simple shapes into other useful shapes.

What is a linear fractional transformation?

A linear fractional transformation has the form

$$f(z)=\frac{az+b}{cz+d},$$

where $a,b,c,d$ are complex numbers and $ad-bc\neq 0$.

This condition $ad-bc\neq 0$ is important because it ensures the function is not degenerate. If $ad-bc=0$, the numerator and denominator are linked in a way that can make the mapping collapse into something less interesting.

The name “linear fractional” comes from the fact that both the numerator and denominator are linear expressions in $z$. These transformations are also called Möbius transformations, after the mathematician August Ferdinand Möbius.

A simple example is

$$f(z)=\frac{z-1}{z+1}.$$

Here, $a=1$, $b=-1$, $c=1$, and $d=1$. Since $ad-bc=1-(-1)=2\neq 0$, this is a valid linear fractional transformation.

These functions are used everywhere in complex analysis because they are powerful, flexible, and mathematically elegant. They can send lines to circles, circles to lines, and many regions into easier ones to study.

Why are they so important in conformal mapping?

A mapping is conformal if it preserves angles between curves at points where the derivative is nonzero. Linear fractional transformations are conformal at every point where they are defined and have nonzero derivative.

For

$$f(z)=\frac{az+b}{cz+d},$$

the derivative is

$$f'(z)=\frac{ad-bc}{(cz+d)^2}.$$

Because $ad-bc\neq 0$, the derivative is nonzero whenever $cz+d\neq 0$. That means the mapping is conformal everywhere except at the point where $cz+d=0$, which is

$$z=-\frac{d}{c}$$

when $c\neq 0$.

This is a major reason these transformations matter. They give examples of functions that preserve local angles, making them useful for reshaping regions without breaking the local geometry. In physical applications, such as fluid flow or electrostatics, this can simplify difficult shapes into manageable ones.

For example, a complicated region in the $z$-plane might be transformed into the upper half-plane or the unit disk, where calculations are easier. Then results can be translated back to the original region.

Main properties and building blocks

A useful way to understand linear fractional transformations is to see them as combinations of simpler maps.

They can be built from these basic transformations:

translation: $z\mapsto z+b$
scaling and rotation: $z\mapsto az$
inversion: $z\mapsto \frac{1}{z}$

By combining these, many Möbius transformations can be created. This is one reason they are so versatile.

Another key idea is that they act on the extended complex plane, also called the Riemann sphere, which includes the point at infinity, $\infty$. This extension is necessary because a linear fractional transformation may send a finite point to $\infty$, or $\infty$ to a finite point.

For example, in

$$f(z)=\frac{1}{z},$$

we have $f(0)=\infty$ and $f(\infty)=0$.

More generally, if $c\neq 0$, then the point where $cz+d=0$ is mapped to $\infty$. Also, as $z\to \infty$,

$$f(z)=\frac{az+b}{cz+d}\to \frac{a}{c}$$

when $c\neq 0$.

If $c=0$, then the formula becomes

$$f(z)=\frac{az+b}{d},$$

which is just an affine transformation, meaning a combination of scaling, rotation, and translation.

How lines and circles are mapped

One of the most famous facts about linear fractional transformations is that they map generalized circles to generalized circles. A generalized circle means either a circle or a straight line, where a line is viewed as a circle passing through $\infty$.

This is a huge geometric feature. If you know where three distinct points go, then a Möbius transformation is essentially determined, because generalized circles are determined by such point behavior.

Let’s look at a useful example:

$$f(z)=\frac{z-i}{z+i}.$$

This transformation maps the real axis to the unit circle. Why? If $z=x$ is real, then

$$|f(x)|=\left|\frac{x-i}{x+i}\right|=1,$$

because $|x-i|=|x+i|$ for real $x$. So every real number is sent to a point on the unit circle.

It also gives a nice mapping of the upper half-plane to the unit disk. If $\operatorname{Im}(z)>0$, then

$$\left|\frac{z-i}{z+i}\right|<1,$$

so the image lies inside the unit circle. This is a classic example of how linear fractional transformations help convert one region into another region that is easier to study.

Another common example is

$$f(z)=\frac{1+z}{1-z},$$

which maps the unit disk to a half-plane. These examples show how Möbius transformations can connect different standard regions in complex analysis.

Working through a concrete transformation

Consider the transformation

$$f(z)=\frac{2z+3}{z-4}.$$

Here $a=2$, $b=3$, $c=1$, and $d=-4$. The determinant condition gives

$$ad-bc=2(-4)-3(1)=-11\neq 0,$$

so this is a valid linear fractional transformation.

To understand its behavior, check a few special points:

At $z=0$,

$$f(0)=\frac{3}{-4}=-\frac{3}{4}.$$

At $z=4$, the denominator is $0$, so

$$f(4)=\infty.$$

As $z\to\infty$,

$$f(z)\to \frac{2}{1}=2.$$

So this map sends the point $4$ to infinity and sends infinity to $2$.

Now think about a line or circle in the $z$-plane. Under this transformation, its image will again be a line or circle. That makes the map useful for changing the shape of a region while keeping its broad geometric type.

If you wanted to study a region with a boundary line passing through $z=4$, this transformation could move that troublesome point to infinity, simplifying the picture. That is a typical conformal mapping strategy.

Composition and inverse transformations

Linear fractional transformations are closed under composition. That means if $f(z)$ and $g(z)$ are both Möbius transformations, then so is $f(g(z))$.

This property is important because many complicated conformal maps are built by combining simpler ones. The inverse of a linear fractional transformation is also a linear fractional transformation, provided the transformation is one-to-one on the extended complex plane.

$$w=\frac{az+b}{cz+d},$$

then solving for $z$ gives

$$z=\frac{dw-b}{-cw+a},$$

assuming $ad-bc\neq 0$. So the inverse also has the same form.

This means Möbius transformations form a very structured family. They do not just work one way; they can be undone. That is a major advantage in mapping problems because it allows you to move to a simpler domain and then return to the original domain.

Connection to the broader topic of conformal mapping

students, conformal mapping is about preserving local angle structure while transforming regions. Linear fractional transformations are among the cleanest and most useful examples of this idea.

They matter in the broader topic because they:

preserve angles where the derivative is nonzero
map circles and lines to circles and lines
move standard regions like half-planes and disks into one another
act naturally on the extended complex plane
can be combined and reversed easily

These properties make them a foundation for more advanced conformal maps. Even when a problem uses a more complicated function, the strategy often starts with a linear fractional transformation to simplify the geometry.

For instance, a difficult boundary might first be sent to a line or circle, then another map may be applied to reach the final desired region. In this way, linear fractional transformations act like a first translation step in a geometric puzzle 🧩.

Conclusion

Linear fractional transformations are a central tool in complex analysis because they are both algebraically simple and geometrically powerful. A map of the form

$$f(z)=\frac{az+b}{cz+d}, \qquad ad-bc\neq 0,$$

is conformal wherever it is defined, and it sends generalized circles to generalized circles. It can move lines to circles, disks to half-planes, and finite points to infinity.

For conformal mapping, this means students can use these transformations to reshape regions while keeping local angles intact. They are one of the clearest examples of how complex functions can transform geometry in elegant and useful ways.

Study Notes

A linear fractional transformation has the form $f(z)=\frac{az+b}{cz+d}$ with $ad-bc\neq 0$.
It is also called a Möbius transformation.
Its derivative is $f'(z)=\frac{ad-bc}{(cz+d)^2}$, so it is conformal where defined and where $f'(z)\neq 0$.
It acts on the extended complex plane, including the point $\infty$.
It maps circles and lines to circles and lines.
Important special cases include $f(z)=\frac{1}{z}$ and $f(z)=\frac{z-i}{z+i}$.
The real axis can map to the unit circle, and the upper half-plane can map to the unit disk.
The inverse of a linear fractional transformation is also a linear fractional transformation.
These maps are useful because they simplify regions while preserving angles locally.