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.
The solution \(G\) together with its inverse function \(F=G^{-1}\) allow to express the non-integer iterates of the transfer function \(T\). The examples of these iterates are considered in book «Superfunctions» , 2020. The main claim is that for any growing real-holomorphic transfer function \(T\), the corresponding real-holomorphic solution \(G\) and its inverse \(F=G^{-1}\) can be constructed for the Abel equation.

The Abel equation appears at the top of series of the conjugations that include
the Abel equation,
the Schroeder equation,
the Boettcher equation and
other similar equations that have not yet established names.

The Abel equation is named after Niels Henrik Abel (see picture below) and his paper «Une équation d’un degré quelconque étant proposée, reconnaître si elle pourra être satisfaite algébriquement, ou non.», 1881. To year 2026, no free online version of this paper is found.

Niels Henrik Abel

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 \(F\); at least in some part of the complex plane, \(F=G^{-1}\) and \(G=F^{-1}\).

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

\((3)~ ~ ~ ~ ~ T^n(z)=F(n+G(z))\)

which, at least for some values of \(z\), satisfies relation \(T^{n+m}(z) = T^n(T^m(z))\); in general, parameters \(n\) and \(m\) 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.

To year 2026, Wikipedia makes no difference between the Abel equation (1) and the Transfer equation (2).

Sometimes it is useful, to have different names for the equation (1) and equation (2), and different names for their solutions. In TORI,
equation (1) is called Abel equation, and its solution is called Abelfunction, and
equation (2) is called Transfer equation (although term Transfer equation may have also other meaning(s)), and its solution is called Superfunction.

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 \(n\!=\!1\).

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.

Conjugations

The Abel equation appears at the top of series of the conjugations that include
the Abel equation \(\ G(T(z))=G(z)+1 \ \) ,
the Schroeder equation \(\ G(T(z))=s\, G(z) \ \) ,
the Boettcher equation \(\ G(T(z))= G(z)^k\)
and other similar equations that have not yet established names, for example,
the Tori equation \(\ G(T(z))= \exp_b(G(z))\ \), and, in general, for arbitrary function \(\Phi\)
the Phi equation \(\ G(T(z))= \Phi(G(z))\ \).

Warning

This article is uploaded at TORI in order to systematize the notations used in book «Superfunctions» .

Editor tries to follow the commonly used notations, but still, the interpretation suggested may deviate from those of other sites.

References

1998.oo.oo. http://matwbn.icm.edu.pl/ksiazki/sm/sm127/sm12716.pdf G.Belitskii, Yu.Lubich. The Abel equations and total solvability of linear functional equations. Studia Maghematica v.127 (1), 1998, p.81-97

2014.08.19. https://jbonet.webs.upv.es/wp-content/uploads/2016/05/Bonet_Domanski.pdf Jos ́e Bonet and Pawel􏰀 Doman ́ski. Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integr. Equ. Oper. Theory 81 (2015), 455–482 DOI 10.1007/s00020-014-2175-4. Published online August 19, 2014.

2015.04.30. http://jbonet.webs.upv.es/wp-content/uploads/2014/04/BD_eigenvaluessubmitted03032014.pdf Jose Bonet, Pawel Domanski. Abel’s Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integral Equations and Operator Theory, April 2015, Volume 81, Issue 4, pp 455–482.

Keywords

«Abel equation», «Abel function», «Abelfunction», «Asymptotic», «Hellmuth Kneser», «Iterate», «Niels Henrik Abel», «Regular iteration», «Superfunction», «Superfunctions», «Transfer equation», «Transfer function»,