Linear fraction

Linear fraction is meromorphic function that can be expressed with

$(1) ~ ~ ~ \displaystyle T(z)=\frac{u+v z}{w+z}$

where $u$, $v$, $w$ are parameters from the some set of numbers that allows operations of summation, multiplication and division. Usually, it is assumed, that they are complex numbers, and the operation of multiplication is commutative.

Linear function

Definition (1) excludes the case of linear function. However, this the linear function can be realized in limit

$(2) ~ ~ ~ \displaystyle A+B z= \lim_{M\rightarrow \infty} \frac{M A+ M B}{M+z}$

where the expression of the function under the limit operation is expressed in a form that corresponds to (1), id est, $u=M A$, $v=MB$, $w=M$.

Inverse function

The inverse function $T^{-1}$ of the linear fraction $T$ by (1) is also linear fraction, and its parameters can be easy expressed through the parameters of the initial linear fraction.

$(3) ~ ~ ~ \displaystyle T^{-1}(z)=\frac{u-w z}{-v+z}$

One can easy check that $T(T^{-1}(z))=T^{-1}(T(z))=z$ for all $z$ excluding the poles, singularities at $z=-w$ and at $z=v$.

Linear conjugate of linear fraction

Linear conjugate of a function $T$ is function $Q\circ T\circ P$ where $P$ is linear function and $Q=P^{-1}$.

The linear function $P$ can be parametrized with two parameters, $A$ and $B$, as follows:

$(4) ~ ~ ~ \displaystyle P(z)=A+B z$

then

$(5) ~ ~ ~ \displaystyle Q(z)=(z-A)/B$

and

$(6) ~ ~ ~ \displaystyle Q \circ T \circ P(z)= \frac{\frac{-A^2+A v-A w+u}{B^2}+\frac{B v-A B}{B^2} z}{\frac{A+w}{B}+z}$

Iterate of linear fraction

Iterate of linear fraction can be expressed in a closed form even for a non-integer number of iteration through its superfunction and the Abel function; as usually, the additional conditions on the asymptotic behavior of these functions is required in order to make the non-integer iterate unique.

References

http://mathworld.wolfram.com/LinearFractionalTransformation.html

http://www.mathpages.com/home/kmath464/kmath464.htm