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

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

This form, at special choice of parameters $A$ and $B$ can be used in order to simplify the construction of non-integer iterates of the linear fraction. One of possible choices is

$(7) ~ ~ ~ \displaystyle \frac{v-A}{B}=1 ~ ~ $, $~ ~ ~ \displaystyle -A^2+Av-Aw+u=0$

and then

$(8) ~ ~ ~ \displaystyle A=\frac{v-w}{2}+r$

$(9) ~ ~ ~ \displaystyle B=v-A=\frac{v+w}{2}-r$

where

$(10) ~ ~ ~ \displaystyle r=\sqrt{\Big(\frac{v-w}{2}\Big)^2+u}$

With such a choice,

$(11) ~ ~ ~ \displaystyle t(z)=Q \circ T \circ P(z)= \frac{z}{c+z}$

where

$(12) ~ ~ ~ \displaystyle c=\frac{v+w+2r}{v+w-2r}$

Then, the linear fraction $T$ by (1) can be written as

$(13) ~ ~ ~ \displaystyle T(z)=P \circ t \circ Q(z)$

and the $n$th iterate of $T$ can be written as

$(14) ~ ~ ~ \displaystyle T^n(z)=P \circ t^n \circ Q(z)$

Iterate of linear fraction

Non–integer Iterate of the linear fraction or the special form $t$ by ( 11) is well solvable;

$(15) ~ ~ ~ \displaystyle t^n(z)=\frac{z}{c^n+\frac{1-c^n}{1-c} z }$

At fixed value of $z\mapsto z_0=\mathrm const$, this expression can be considered a function of number of iteration $n$, giving the expression for the superfunction $f$

$(16) ~ ~ ~ \displaystyle f(z)= \frac{z_0}{c^z+\frac{1-c^z}{1-c} z_0 }= \frac{(1-c)z_0}{(1-c) c^z+z_0-z_0 c^z }= \frac{(1-c)z_0}{(1-c-z_0) c^z+z_0 } $

constant $z_0$ should be chosen in a way to simplify the expression (16).

Choice $z_0=-c$ gives superfunction

$(17) ~ ~ ~ \displaystyle f(z)= \frac{- (1-c) c}{c^z-c }= \frac{c\!-\!1}{c^{z-1}-1 }$ that cannot be used for evaluation of the first iterate, but choice $z_0=1$ gives

Especially simple the interpretation above is for the case $r>0$, then, the initial function $T$ by (1) has at elast one fixed point, and the $n$th iteration, id est, $t^n$ is regular at this fixed point even at non-integer values of $n$; the iteration also can be expressed as a linear fraction.

Superfunction of the linear fraction

The Iterate of linear fraction can be expressed 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.

Keywords

Iterate of function, Superfunction, Holomorphic function

References

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

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