Superfunction: Difference between revisions
imported>Dmitrii Kouznetsov m (→Rational function: misprint) |
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===Algebraic function=== | ===Algebraic function=== | ||
In the similar way one can consider the transfer function | |||
:<math>H(z)=2z \sqrt{1-z^2}</math> | |||
and | |||
:<math>F(z)=\sin(\pi 2^z)</math> | |||
which is <math>(C,~ 0\!\rightarrow \!0)</math> superfunction of <math>H</math> for | |||
<math>C= \{z\in \mathbb C : \Re( \cos(\pi 2^z))>0 \}</math>. | |||
===Exponentiation=== | ===Exponentiation=== |
Revision as of 18:57, 28 April 2009
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Superfunction comes from iteration of another function. Roughly, for some function and for some constant , the superfunction could be defined with expression
then can be interpreted as superfunction of function . Such definition is valid only for positive integer . The most research and appllications around the superfunctions is related with various extensions of superfunction; and analysis of the existence, uniqueness and ways of the evaluation. For simple function , such as addition of a constant or multiplication by a constant, the superfunction can be expressed in terms of elementary function. In particular, the Ackernann functions and tetration can be interpreted in terms of super-functions.
History
Analysis of superfunctions came from the application to the evaluation of fractional iterations of functions. Superfunctions and their inverse functions allow evaluation of not only minus-first power of a function (inverse function), but also any real and even complex iteration of the function. Historically, the first function of such kind considered was ; then, function was used as logo of the Physics department of the Moscow State University [1][2][3]. That time, researchers did not have computational facilities for evaluation of such functions, but the was more lucky than the ; at least the existence of holomorphic function such that has been demonstrated in 1950 by Helmuth Kneser [4]. Actually, for his proof, Kneser had constructed the superfunction of exp and corresponding Abel function , satisfying the Abel equation
the inverse function is the entire super-exponential, although it is not real at the real axis; it cannot be interpreted as tetration, because the condition cannot be realized for the entire super-exponential.
Extensions
The recurrence above can be written as equations
- .
Instead of the last equation, one could write
and extend the range of definition of superfunction to the non-negative integers. Then, one may postulate
and extend the range of validity to the integer values larger than . The following extension, for example,
is not trifial, because the inverse function may happen to be not defined for some values of . In particular, tetration can be interpreted as super-function of exponential for some real base ; in this case,
then, at ,
- .
but
- .
For extension to non-integer values of the argument, superfunction should be defined in different way.
Definition
For complex numbers and , such that belongs to some connected domain ,
superfunction (from to ) of holomorphic function on domain is
function , holomorphic on domain , such that
- .
Uniqueness
In general, the super-function is not unique. For a given base function , from given superfunciton , another super-function could be constructed as
where is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that .
The modified super-function may have narrowed range of holomorphism. The variety of possible super-functions is especially large in the limiting case, when the width of the range of holomorphizm becomes zero; in this case, one deals with the real-analytic superfunctions [5].
If the range of holomorphism required is large enough, then, the super-function is expected to be unique, at least in some specific base functions . In particular, the super-function of , for , is called tetration and is believed to be unique at least for ; for the case , see [6]; but up to year 2009, the uniqueness is rather conjecture than a theorem with the formal mathematical proof.
Examples
Addition
Chose a complex number and define function with relation . Define function with relation .
Then, function is superfunction ( to ) of function on .
Multiplication
Exponentiation is superfunction (from 1 to ) of function .
Quadratic polynomials
Let . Then, is a superfunction of .
Indeed,
and
In this case, the superfunction is periodic; its period
and the superfunction approaches unity also in the negative direction of the real axis,
The example above and the two examples below are suggested at [7]
Rational function
In general, the transfer function has no need to be entire function. Here is the example with meromorphic function . Let
- ;
Then, function
is superfunction of function , where is the set of complex numbers except singularities of function . For the proof, the trigonometric formula
can be used at , that gives
Algebraic function
In the similar way one can consider the transfer function
and
which is superfunction of for .
Exponentiation
Let , , . Then, tetration is a superfunction of .
Abel function
Inverse of superfunction can be interpreted as the Abel function.
For some domain and some ,,
Abel function (from to ) of function with respect to superfunction on domain
is holomorphic function from to such that
The definitionm above does not reuqire that ; although, from properties of holomorphic functions, there should exist some subset such that . In this subset, the Abel function satisfies the Abel equation.
Abel equation
The Abel equation is some equivalent of the recurrent equation
in the definition of the superfunction. However, it may hold for from the reduced domain .
Applications of superfunctions and Abel functions
References
- ↑ Logo of the Physics Department of the Moscow State University. (In Russian); http://zhurnal.lib.ru/img/g/garik/dubinushka/index.shtml
- ↑
V.P.Kandidov. About the time and myself. (In Russian)
http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf:
По итогам студенческого голосования победителями оказались значок с изображением
рычага, поднимающего Землю, и нынешний с хорошо известной эмблемой в виде корня из факториала, вписанными в букву Ф. Этот значок, созданный студентом кафедры биофизики А.Сарвазяном, привлекал своей простотой и выразительностью. Тогда эмблема этого значка подверглась жесткой критике со стороны руководства факультета, поскольку она не имеет физического смысла, математически абсурдна и идеологически бессодержательна.
- ↑
250 anniversary of the Moscow State University. (In Russian)
ПЕРВОМУ УНИВЕРСИТЕТУ СТРАНЫ - 250!
http://nauka.relis.ru/11/0412/11412002.htm
На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее.
- ↑ H.Kneser. “Reelle analytische L¨osungen der Gleichung und verwandter Funktionalgleichungen”. Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.
- ↑ P.Walker. Infinitely differentiable generalized logarithmic and exponential functions. Mathematics of computation, 196 (1991), 723-733.
- ↑ D.Kouznetsov. Analytic solution of F(z+1)=exp(F(z)) in complex z-plane. Mathermatics of Computation, 2009, http://www.ams.org/mcom/0000-000-00/S0025-5718-09-02188-7/home.html
- ↑ Mueller. Problems in Mathematics. http://www.math.tu-berlin.de/~mueller/projects.html