Difference between revisions of "Abel equation"

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'''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:
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'''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:
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: $(1)~ ~ ~ ~ ~ G(T(z))=G(z)+1$
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: \((1)~ ~ ~ ~ ~ G(T(z))=G(z)+1\)
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at least for $z$ from some domain in the complex plane.
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at least for \(z\) from some domain in the complex plane.
   
 
===Transfer equation===
  
===Transfer equation===
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The Abel equation is closely related to the [[transfer equation]] for the [[superfunction]] $F$:
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The Abel equation is closely related to the [[transfer equation]] for the [[superfunction]] \(F\):
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: $ (2)~ ~ ~ ~ ~ F(z\!+\!1)=T(F(z))$
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: \( (2)~ ~ ~ ~ ~ F(z\!+\!1)=T(F(z))\)
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The [[Abel function]] $G$ is considered as inverse of the [[superfunction]] $F$.
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The [[Abel function]] \(G\) is considered as inverse of the [[superfunction]] \(F\).
   
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Together, the [[Abel function]] and the [[superfunction]] allow to express the $c$th [[iteration]] of the [[transfer function]] $h$ as follows:
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Together, the [[Abel function]] and the [[superfunction]] allow to express the \(c\)th [[iteration]] of the [[transfer function]] \(h\) as follows:
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: $(3)~ ~ ~ ~ ~ T^c(z)=F(c+G(z))$
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: \((3)~ ~ ~ ~ ~ T^c(z)=F(c+G(z))\)
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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 $T~, ~ ~$
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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 \(T~, ~ ~\)
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$T^0$ is identity function, $T^1\!=\!T$ and so on.
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\(T^0\) is identity function, \(T^1\!=\!T\) and so on.
   
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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.
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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===
  
===Uniqueness===
   
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For the transfer function $T$ of general kind, the problem of existence and uniqueness of solution of the [[Abel equation]] is not trivial.
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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 function]]s, and the corresponding [[Abel function]] can be constructed;
  
Most of commonly used functions can be declared as [[transfer function]]s, 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
  
better to say, many of them can be constructed. The additional conditions, for example, the asymptotic the infinity and
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<ref name="kneser">
  
<ref name="kneser">
 
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 [[H.Kneser]]. Reelle analytische Lösungen der Gleichung
 
http://www.digizeitschriften.de/dms/img/?PPN=GDZPPN002175851 [[H.Kneser]]. Reelle analytische Lösungen der Gleichung
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$\varphi(\varphi(x))=e^x$. [[Equationes Mathematicae]] (Journal fur die reine und angewandte Mathematik) '''187''' 56–67 (1950)
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\(\varphi(\varphi(x))=e^x\). [[Equationes Mathematicae]] (Journal fur die reine und angewandte Mathematik) '''187''' 56–67 (1950)
 
</ref><ref name="uniabel">
  
</ref><ref name="uniabel">
 
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)
  
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)
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===Examples===
  
===Examples===
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The [[Abel equation]] becomes simple, if the [[transfer function]] $T$ is considered as unknown, while the
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The [[Abel equation]] becomes simple, if the [[transfer function]] \(T\) is considered as unknown, while the
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[[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:
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[[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:
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: $(4)~ ~ ~ ~ ~ T(z)=F(1+G(z))$
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: \((4)~ ~ ~ ~ ~ T(z)=F(1+G(z))\)
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Actually, such an expression is just a special case of equation (3) for $c\!=\!1$.
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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,
 
The representation (4) allows to construct many examples. One can see that the [[division]] by a constant is [[Abel function]] of addition,

Latest revision as of 18:26, 30 July 2019

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.

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 [1][2].

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.

References

  1. ↑ 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)
  2. ↑ 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)

2014.08.19. http://download.springer.com/static/pdf/913/art%253A10.1007%252Fs00020-014-2175-4.pdf?originUrl=http%3A%2F%2Flink.springer.com%2Farticle%2F10.1007%2Fs00020-014-2175-4&token2=exp=1452862099~acl=%2Fstatic%2Fpdf%2F913%2Fart%25253A10.1007%25252Fs00020-014-2175-4.pdf%3ForiginUrl%3Dhttp%253A%252F%252Flink.springer.com%252Farticle%252F10.1007%252Fs00020-014-2175-4*~hmac=f1fad8b300e8acbc3efd12bc68adde6dbc28b51cb981bfe01aa64dc9f2ab3c64 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.

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