# Transfer equation

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

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

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

## 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 $$h$$.

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 $$h$$ 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 $$h$$ and $$F$$ have values from the set of complex numbers, the problem of finding of superfunction $$F$$ for given transfer function $$h$$ 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

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

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

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

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

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

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

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

## Examples

Several examples of pairs (superfunction, Abel function) are suggested in the Moscow University Physics Bylletin [1]. 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 [2].

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.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.
2. 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)