# Transfer equation

Transfer equation is relation between some holomorphic function $T$, called transfer function, and another function $F$, called Superfunction, expressed with

$F(z\!+\!1)=T(F(z))$

for all $z\in C \subseteq \mathbb C$.

Often, the range $C$ cover the most of the complex plane $\mathbb C$, especially for entire transfer function $T$.

## Physical sense of the transfer equation

The transfer equation describes the evolution of some isolated system in the discrete time, when the state in the next moment o time is completely determined by its state in the previous moment of time; this determinism is expressed with the transfer function $T$.

In principle $F(z)$ may belong some Hilbert space; then the transfer equation may correspond to the evolution of the state of a system in the discrete time $z$; then $T$ is analogy of the Hamiltonian. The extension to the complex, and, in particular, to the real values of time $z$ refers to the conventional Quantum mechanics, where time appears as real parameter.

It happened, that even in the case of simple, "zero-dimensional" Hilbert space, when $T$ and $F$ have values from the set of complex numbers, the problem of finding of superfunction $F$ for given transfer function $T$ is not trivial. This determined the interest to the superfunctions from the set of holomorphic functions of complex variable. The extension to the complex values of $z$ happens to be essential for the establishment of the unique superfunctions for some simple transfer dunctions auch as exponential or the logistic operator.

## Abel function and Abel equation

The Inverse of the Superfunction, $G=F^{-1}$ is called Abel function. Within some domain $D\in \mathbb C$, the Abel function satisfies the Abel equation

$G(T(z))=G(z)+1$

The Abel function, together with the corresponding superfunction $F$, allows to express the iteration of the transfer function $h$ in the following form:

$T^c(z)=F(c+G(z))$

for all z from some domain in the set of complex numbers, where the number $c$ of iterations have no need to be integer; and the relation

$T^c(T^d(z))=T^{c+d}(z)$

for all complex $c,d,z$ from some domain which may cover significant part of the complex plane.

In particular, $T^{-1}$ is the inverse function, $T^0$ is the identity function and $T^1\!=\!T$.

## History

Since century 20, the transfer equation had been considered mainly for construction of non-integer iterates of exponential. [1][2]

Then, non–trivial iterates of other functions had been considered.

## Examples

Several examples of pairs (superfunction, Abel function) are suggested in the Moscow University Physics Bulletin [3]. In particular, the tetration satisfies the transfer equation with the transfer function exp.

The superfunction and its inverse for the subsequence of the Collatz sequence are considered in the article [4].

The superfunction for the holomorphic extension of the second iteration of the Optimized Collatz operator are considered in the article Holomorphic_extension_of_the_Collatz_subsequence.

## References

1. http://www.ams.org/journals/mcom/2009-78-267/S0025-5718-09-02188-7/S0025-5718-09-02188-7.pdf D. Kouznetsov (July 2009). "Solution of F(z+1)=exp(F(z)) in complex z-plane". Mathematics of Computation. 78 (267): 1647–1670.
2. http://myweb.astate.edu/wpaulsen/tetration2.pdf William Paulsen and Samuel Cowgill. Solving F (z+1) = b^F(z) in the complex plane. Advances in Computational Mathematics, 2017, March, p. 1–22.
3. http://www.springerlink.com/content/qt31671237421111/fulltext.pdf?page=1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
4. http://www.springerlink.com/content/u712vtp4122544x4 D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98. (Russian version: p.24-31)