Abel equation

Abel equation is functional equation that relates some known function (considered as transfer function) $$T$$ to the corresponding Abel function $$G$$ in the following way:

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

at least for $$z$$ from some domain in the complex plane.

Contents

Transfer equation

The Abel equation is closely related to the transfer equation for the superfunction $$F$$:

$$(2)~ ~ ~ ~ ~ F(z\!+\!1)=T(F(z))$$

The Abel function $$G$$ is considered as inverse of the superfunction $$F$$.

Together, the Abel function and the superfunction allow to express the $$c$$th iteration of the transfer function $$h$$ as follows:

$$(3)~ ~ ~ ~ ~ T^c(z)=F(c+G(z))$$

which, at least for some values of $$z$$, satisfies relation $$T^{c+d}(z) = T^c(h^d(z))$$; in general, parameters $$c$$ and $$d$$ have no need to be integer. For the case of integer iterations, $$T^{-1}$$ is inverse function of $$T~, ~ ~$$ $$T^0$$ is identity function, $$T^1\!=\!T$$ and so on.

Once the Superfunction $$F$$ and the corresponding Abel function $$G$$ are specified, the transfer function $$T$$ by (3) can be easily iterated arbitrary number of times, in particular, non-integer and even complex iteration is available.

Uniqueness

For the transfer function $$T$$ of general kind, the problem of existence and uniqueness of solution of the Abel equation is not trivial. Most of commonly used functions can be declared as transfer functions, and the corresponding Abel function can be constructed; better to say, many of them can be constructed. The additional conditions, for example, the asymptotic the infinity and the behavior in vicinity of the fixed points can be used to specify the unique solution .

Examples

The Abel equation becomes simple, if the transfer function $$T$$ is considered as unknown, while the Transfer function $$F$$ and its invese, id est the Abel function $$G$$, are given. (in general any non-trivial function has many inverse functions). Then, the transfer function $$T$$ can be expressed as follows:

$$(4)~ ~ ~ ~ ~ T(z)=F(1+G(z))$$

Actually, such an expression is just a special case of equation (3) for $$c\!=\!1$$.

The representation (4) allows to construct many examples. One can see that the division by a constant is Abel function of addition, logarithm is Abel function of addition and so on.

More examples are considered in article Transfer function.