# LinearFractionalTransformations

This module defines a `LFT`

data type to represent a complex *linear
fractional transformation*. This is a function on the extended
complex numbers (include complex infinity) defined by

```
f(z) = (a*z + b) / (c*z + d)
```

where `a,b,c,d`

are (finite) complex numbers and `a*d-b*c != 0`

.

These are also known as *Möbius transformations*.

## Constructors

The basic constructor takes four values:

```
julia> using LinearFractionalTransformations
julia> julia> f = LFT(1,2,3,4)
LFT( 1.0 + 0.0im , 2.0 + 0.0im , 3.0 + 0.0im , 4.0 + 0.0im )
```

Notice that the `LFT`

is represented by a 2-by-2 complex matrix.
A `LFT`

can also be defined by specifying a matrix.

```
julia> A = [1 2; 3 4];
julia> g = LFT(A)
LFT( 1.0 + 0.0im , 2.0 + 0.0im , 3.0 + 0.0im , 4.0 + 0.0im )
```

The identity `LFT`

is constructed by `LFT()`

:

```
julia> LFT()
LFT( 1.0 + 0.0im , 0.0 + 0.0im , 0.0 + 0.0im , 1.0 + 0.0im )
```

Given (complex) numbers `a,b,c`

(including `Inf`

) we can construct
a `LFT`

that maps `a`

to 0, `b`

to 1, and `c`

to infinity.

```
julia> f = LFT(2,5,-1)
LFT( 6.0 + 0.0im , -12.0 + 0.0im , 3.0 + 0.0im , 3.0 + 0.0im )
julia> f[2]
0.0 + 0.0im
julia> f[5]
1.0 + 0.0im
julia> f[-1]
Inf + Inf*im
```

Finally, we provide a constructor for mapping a given triple of values
`(a,b,c)`

to another triple `(aa,bb,cc)`

. The syntax is
`LFT(a,aa,b,bb,c,cc)`

. Here's an example:

```
julia> f = LFT(1,2+im, 3,Inf, 4,1-im)
LFT( 5.0 + 1.0im , -17.0 - 7.0im , 3.0 + 0.0im , -9.0 + 0.0im )
julia> f[1]
2.0 + 1.0im
julia> f[3]
Inf + Inf*im
julia> f[4]
1.0 - 1.0im
```

#### Under the hood

The matrix representing a `LFT`

object is held in a field named `:M`

.

```
julia> f = LFT(1,2,3)
LFT( -1.0 + 0.0im , 1.0 + 0.0im , 1.0 + 0.0im , -3.0 + 0.0im )
julia> f.M
2x2 Array{Complex{Float64},2}:
-1.0+0.0im 1.0+0.0im
1.0+0.0im -3.0+0.0im
```

## Operations

### Function application

Since a `LFT`

is a function, the most basic operation we may wish to
perform is applying that function of a complex number. That's done
with `f[x]`

notation (or with `f(x)`

):

```
julia> f = LFT(3,2,1,1)
LFT( 3.0 + 0.0im , 2.0 + 0.0im , 1.0 + 0.0im , 1.0 + 0.0im )
julia> f[1]
2.5 + 0.0im
julia> f[0]
2.0 + 0.0im
julia> f[-1]
Inf + Inf*im
julia> f[Inf]
3.0 + 0.0im
julia> f[1+2im]
2.75 + 0.25im
```

**Note**: Staring in Julia 0.4, I plan to replace `f[x]`

with `f(x)`

by defining `call`

.

### Composition and inverse

The `*`

operation is used for function composition.

```
julia> f = LFT(3,2,1,1);
julia> g = LFT(0,1,-1,2);
julia> f*g
LFT( -2.0 + 0.0im , 7.0 + 0.0im , -1.0 + 0.0im , 3.0 + 0.0im )
julia> g*f
LFT( 1.0 + 0.0im , 1.0 + 0.0im , -1.0 + 0.0im , 0.0 + 0.0im )
```

The inverse of a `LFT`

is computed with `inv`

:

```
julia> f = LFT(1,2,3,4);
julia> g = inv(f)
LFT( 4.0 + 0.0im , -2.0 - 0.0im , -3.0 - 0.0im , 1.0 + 0.0im )
julia> f*g
LFT( -2.0 + 0.0im , 0.0 + 0.0im , 0.0 + 0.0im , -2.0 + 0.0im )
```

Notice that the matrix representing `f*g`

is a scaled version of the
identity matrix.

## Equality checking

We can use `==`

or `isequal`

to check if two `LFT`

objects are
equal. Note that there is no unique matrix representation for a `LFT`

object and we might have that `f`

and `g`

are equal, but `f.M`

and
`g.M`

are different.

```
julia> f = LFT(1,2,3,4);
julia> g = LFT(-2,-4,-6,-8);
julia> f==g
true
julia> f.M == g.M
false
```