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:
- \((1)~ ~ ~ ~ ~ G(T(z))=G(z)+1\)
for \(z\) from some domain \(D\) in the complex plane. It is assumed that \(T\) is holomorphic (or real-analytic) function defined on a domain \(D\subset\mathbb C\). Often it is assumed also that along the real axis, \(T\) is growing function.
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 the transfer functions \(T\) and the solutions \(G\) are considered in book «Superfunctions» [1], 2020, although no general proof of existence of the real-holomorphic Abel functions \(G\) for arbitrary growing real-holomorphic transfer function \(T\) is provided.
The Abel equation appears as a simplest equation in the set or its conjugations; his set includes also 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). He seems to mention [2] the topic in 1881. It is difficult to verity: to year 2026, no free online version of paper [2] is found.
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. It is assumed that both \(F\) and \(G\) are analytic in suitable domains, and the inversion is well-defined. In general, TORI do not deal with multivalued functions; so, the branch cut dividing the range of holomophism may destroy the harmony above. The branch cuts are unavoidable, as only in a trivial case both \(G\) and \(F=G^{-1}\) may be entire functions; Usually, at least one of functions \(F\),G\) has branch cuts.
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 [4] 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 [5][6].
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 and in book «Superfunctions»[1].
Conjugations
The Abel equation is one example of conjugating a function to a simpler dynamical model. The described conjugations list: translation, multiplication, power map, but not yet the exponential.
The Abel equation appears among various 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» [1].
Editor tries to follow the commonly used notations, but still, the interpretation suggested may deviate from those of other sites.
The equation is named after Niels Henrik Abel; the terminology originates from later developments in iteration theory.
Acknowledgment
ChatGPT helped to improve this article.
References
- ↑ 1.0 1.1 1.2 https://nizugadro.mydns.jp/BOOK/486.pdf D.Kouznetsov. Superfunctions. Lambert Academic Publishing, 2020.
- ↑ 2.0 2.1 Niels Henrik Abel. “Une équation d’un degré quelconque étant proposée, reconnaître si elle pourra être satisfaite algébriquement, ou non.”. Overs complètes, 1881, vol. 2, 330. (No free online version is found)
- ↑ https://commons.wikimedia.org/wiki/File:Niels_Henrik_Abel.jpg Description Niels Henrik Abel Source Originally uploaded to English wikipedia Painting by Johan Gørbitz (1782–1853)
- ↑ https://en.wikipedia.org/wiki/Abel_equation The Abel equation, named after Niels Henrik Abel, is a type of functional equation of the form \( f(h(x))=h(x+1)\) or \(\alpha (f(x))=\alpha (x)+1\). The forms are equivalent when α is invertible. h or α control the iteration of f.
- ↑ http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 H.Kneser. Reelle analytische Lösungen der Gleichung \(\varphi(\varphi(x))=e^x\). Equationes Mathematicae (Journal fur die reine und angewandte Mathematik) 187 56–67 (1950)
- ↑ http://www.springerlink.com/content/u7327836m2850246/ H.Trappmann, D.Kouznetsov. Uniqueness of Analytic Abel Functions in Absence of a Real Fixed Point. Aequationes Mathematicae, 81, p.65-76 (2011)
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»,