Superfunction: Difference between revisions

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then <math>S</math> can be interpreted as superfunction of function <math>f</math>.
then <math>S</math> can be interpreted as superfunction of function <math>f</math>.
Such definition is valid only for positive integer <math>z</math>.  
Such definition is valid only for positive integer <math>z</math>.  
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.
The most research and applications around the superfunctions are related with various extensions of superfunction; and analysis of the existence, uniqueness and ways of the evaluation.
<!--  In particular, :<math>S(1)=f(t)</math> !-->
<!--  In particular, :<math>S(1)=f(t)</math> !-->
For simple function <math>f</math>, such as addition of a constant or multiplication by a constant,
For simple function <math>f</math>, such as addition of a constant or multiplication by a constant,
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==History==
==History==
[[Image:Sqrt(factorial)LOGOintegralLOGO.jpg|100px|right|thunb|LOGOs of the Phys.Dept. of the MSU and that of Math.Dep.]]
[[Image:Sqrt(factorial)LOGOintegralLOGO.jpg|100px|left|thumb|logos of the Phys.Dept. of the MSU and that of Math.Dep.]]
<!--[[Image:QFacQexp.jpg|right|400px|thumb|<math>\sqrt{!}</math> and <math>\sqrt{\exp}</math> in the complex plane]]
[[Image:QFacQexp.jpg|right|100px]]!-->
[[Image:QFactorialQexp.jpg|256px|thumb|<math>\sqrt{!}</math> and <math>\sqrt{\exp}</math> in the complex plane]]
Analysis of superfunctions came from the application to the evaluation of fractional iterations of functions.  
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 <math>\sqrt{\exp}~</math>; then, function <math>\sqrt{!~}~</math> was used as logo of the Physics department of the [[Moscow State University]]
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 <math>\sqrt{\exp}~</math>; then, function <math>\sqrt{!~}~</math> was used as logo of the Physics department of the [[Moscow State University]]
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На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее.
На значке физфака в букву "Ф" вписано стилизованное изображение корня из факториала (√!) - выражение, математического смысла не имеющее.
</blockquote>
</blockquote>
</ref>.
</ref>. (Mathematicians of the same University were not so arrogant and used the symbol of [[integral]] and the [[Moebius surface]] at their logo, see the figure at left).
 
That time, researchers did not have computational facilities for evaluation of such functions, but
That time, researchers did not have computational facilities for evaluation of such functions, but
the <math>\sqrt{\exp}</math> was more lucky than the <math>~\sqrt{!~}~~</math>; at least the existence of [[holomorphic function]]  
the <math>\sqrt{\exp}</math> was more lucky than the <math>~\sqrt{!~}~~</math>; at least the existence of [[holomorphic function]]  
<math>\varphi</math> such that <math>\varphi(\varphi(z))=\exp(z)</math> has been demonstrated in 1950 by [[Helmuth Kneser]] <ref name="kneser">
<math>\varphi</math> such that <math>\varphi(\varphi(z))=\exp(z)</math> has been demonstrated in 1950 by [[Helmuth Kneser]] <ref name="kneser">
H.Kneser. “Reelle analytische L¨osungen der Gleichung <math>\varphi(\varphi(x)) = e^x </math> und verwandter Funktionalgleichungen”.
{{cite journal
Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.
|author=[[Helmuth Kneser|H.Kneser]]
</ref>. Actually, for his proof, Kneser had constructed the superfunction of exp and corresponding Abel function <math>\mathcal{X}</math>, satisfying the [[Abel equation]]
|title=Reelle analytische L¨osungen der Gleichung <math>\varphi(\varphi(x)) = e^x </math> und verwandter Funktionalgleichungen
: <math>\mathcal{X}(\exp(z))=\mathcal{X}(z)+1</math>
|journal=[[Journal fur die reine und angewandte Mathematik]]
the inverse function <math>F</math> is the [[entire function|entire]] super-exponential, although it is not real at the real axis; it cannot be interpreted as [[tetration]], because the condition <math>F(0)=1</math> cannot be realized for the entire super-exponential.
|volume=187
|year=1950
|pages=56-67}}
</ref>. Actually, for his proof, Kneser had constructed the superfunction of exp and the corresponding Abel function <math>\mathcal{X}</math>, satisfying the [[Abel equation]]
: <math>\mathcal{X}(\exp(z))=\mathcal{X}(z)+1</math> .
The inverse function, id est <math>F=\mathcal \chi^{-1}</math> is an [[entire function|entire]] super-exponential, although it is not real at the real axis; it cannot be interpreted as [[Tetration|tetrational]], because the condition <math>F(0)=1</math> cannot be realized for the entire super-exponential. The [[real function|real]] <math>\sqrt{\exp}</math> can be constructed with the [[tetration|tetrational]] (which is also a superexponential), and the real <math>\sqrt{\rm Factorial}</math> can be constructed with the [[superfactorial]]. The plots of <math>\sqrt{\rm Factorial}</math> and <math>\sqrt{\exp}</math> in the complex plane are shown in the right hand side figure.


==Extensions==
==Extensions==
The recurrence above can be written as equations
The recurrent formula of the preamble can be written as equations
:<math>S(z\!+\!1)=f(S(z)) ~ \forall z\in \mathbb{N} : z>0</math>
:<math>S(z\!+\!1)=f(S(z)) ~ \forall z\in \mathbb{N} : z>0</math>
:<math>S(1)=f(t)</math>.
:<math>S(1)=f(t)</math>.
Line 62: Line 71:
The following extension, for example,
The following extension, for example,
:<math>S(-2)=f^{-2}(t)</math>  
:<math>S(-2)=f^{-2}(t)</math>  
is not trifial, because the inverse function may happen to be not defined for some values of <math>t</math>.
is not trivial, because the inverse function may happen to be not defined for some values of <math>t</math>.
In particular, [[tetration]] can be interpreted as super-function of exponential for some real base <math>b</math>; in this case,
In particular, [[tetration]] can be interpreted as super-function of exponential for some real base <math>b</math>; in this case,
<!-- :<math>f(z)={b}^z</math>!-->
<!-- :<math>f(z)={b}^z</math>!-->
Line 74: Line 83:


==Definition==
==Definition==
For complex numbers <math>~a~</math> and <math>~b~</math>, such that <math>~a~</math> belongs to some domain <math>D\subseteq \mathbb{C}</math>,<br>
For complex numbers <math>~a~</math> and <math>~b~</math>, such that <math>~a~</math> belongs to some connected domain <math>D\subseteq \mathbb{C}</math>,<br>
<!-- <math>a \!\mapsto\! b</math> !-->
<!-- <math>a \!\mapsto\! b</math> !-->
superfunction (from <math>a</math> to <math>b</math>) of [[holomorphic function]] <math>~f~</math> on domain <math>D</math> is
superfunction (from <math>a</math> to <math>b</math>) of [[holomorphic function]] <math>~f~</math> on domain <math>D</math> is
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:<math>S(z\!+\!1)=f(S(z)) ~ \forall z\in D : z\!+\!1 \in D</math>
:<math>S(z\!+\!1)=f(S(z)) ~ \forall z\in D : z\!+\!1 \in D</math>
:<math>S(a)=b</math>.
:<math>S(a)=b</math>.
==Uniqueness==
==Uniqueness==
Holomorphism declared in the definition is essential for the uniqueness. If no additional requirements on the continuity of the function in the complex plane, the strip
In general, the super-function is not unique.
<math> \{ z \in \mathbb{C} : -1<\Re(z)\le 0 \}</math> can be filled with any function
For a given base function <math>H</math>, from given <math>(a\mapsto d)</math> superfunciton <math>F</math>, another <math>(a\mapsto d)</math>  super-function <math>G</math> could be constructed as
(for example, the [[Dirichlet function]]), and extended with the recurrent equation.
: <math>G(z)=F(z+\mu(z))</math>
A little bit more regular approach is fitting of the superfunction at the part of the real axis with some simple function (for example, the linear function), and following extension of this step-vice function to the whole complex plane.
where <math>\mu</math> is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that <math> \mu(a)=0 </math>.
 
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 <ref name="walker">
{{cite journal
|author=P.Walker
|title=Infinitely differentiable generalized logarithmic and exponential functions
|journal= [[Mathematics of computation]]
|volume=196
|year=1991
|pages=723-733
|url=http://www.jstor.org/stable/2938713
}}
</ref>.
 
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 <math>H</math>. In particular, the <math>(C, 0\mapsto 1)</math> super-function of
<math>\exp_b</math>, for <math>b>1</math>, is called [[tetration]] and is believed to be unique at least for
<math>C= \{ z \in \mathbb{C} ~:~\Re(z)>-2 \}</math>; for the case <math> b>\exp(1/\mathrm{e})</math>, see <ref name="kouznetsov">
{{cite journal
|author=D.Kouznetsov.
|title=Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.
|journal=[[Mathematics of Computation]],
|year=2009
|volume=78
|pages=1647-1670
|url= http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
|preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf
|doi=10.1090/S0025-5718-09-02188-7
}}</ref>;
but up to year 2009, the uniqueness is rather [[conjecture]] than a theorem with the formal mathematical proof.


==Examples==
==Examples==
The short table of superfunctions is suggested in
<ref name="superfactorial">
D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. [[Moscow University Physics Bulletin]], 2010, v.65, No.1, p.6-12. (Preprint ILS UEC, 2009:
http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf )
</ref>. A little bit more extended table is available at [[TORI]]
<ref name="toritable">
http://tori.ils.uec.ac.jp/TORI/index.php/Table_of_superfunctions
</ref>. Some superfunctions can be expressed with elementary functions,
they are used without to mention that they are superfunctions.
For example, for the transfer function "++", which means unity increment,
the superfunction is just addition of a constant.
===Addition===
===Addition===
Chose a [[complex number]] <math>c</math> and define function  
Chose a [[complex number]] <math>c</math> and define function  
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<ref name="mueller">Mueller. Problems in Mathematics.
<ref name="mueller">Mueller. Problems in Mathematics.
http://www.math.tu-berlin.de/~mueller/projects.html
http://www.math.tu-berlin.de/~mueller/projects.html
</ref>.
More general case refers to the [[logistic transfer function]] <math> H(z)=u~z~(1-z) </math>
where <math>u>1</math> is parameter; the corresponding superfunction <math>F</math> is called [[logistic sequence]]
<ref name="logistic">
{{cite journal
|url=http://www.springerlink.com/content/u712vtp4122544x4/
|first=Dmitrii
|last=Kouznetsov
|title=Holomorphic extension of the logistic sequence
|journal=[[Moscow University Physics Bulletin]]
|year=2010
|issue=2
|pages=91-98
}}</ref>; <math>F(z\!+\!1)=H(F(z))</math>.
Within wide range of values of parameter <math>u</math>, the logistic sequence <math> H(z)</math> is holomorphic function of <math>z</math>; at <math>u\!=\!4</math>, it can be expressed in a way, equivalent to the representation above <ref name="superfactorial"/>
<!--<ref name="logistic">
{{cite journal
|url=http://www.springerlink.com/content/u712vtp4122544x4/
|first=Dmitrii
|last=Kouznetsov
|title=Holomorphic extension of the logistic sequence
|journal=[[Moscow University Physics Bulletin]]
|year=2010
|issue=2
|pages=91-98
}}
</ref>
</ref>
Should it be cited again?
!-->
.
===Rational function===
===Rational function===
In general, the transfer function <math>H</math> has no  need to be [[entire function]].
In general, the transfer function <math>H</math> has no  need to be [[entire function]].
Here is the example with [[meromorghic function]] <math>H</math>.  
Here is the example with [[meromorphic function]] <math>H</math>.  
Let
Let
:<math>H(z)=\frac{2z}{1-z^2} ~ \forall z\in D~</math>; <math>~ D=\mathbb{C} \backslash  \{-1,1\}</math>
:<math>H(z)=\frac{2z}{1-z^2} ~ \forall z\in D~</math>; <math>~ D=\mathbb{C} \backslash  \{-1,1\}</math>
Line 144: Line 227:


===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===
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The Abel equation is some equivalent of the recurrent equation  
The Abel equation is some equivalent of the recurrent equation  
: <math>F(S(z))=S(z\!+\!1)</math>
: <math>F(S(z))=S(z\!+\!1)</math>
in the definition of the superfunction. However, it may hold for <math>x</math> from the reduced domain <math>\mathcal{D}</math>.
in the definition of the superfunction. However, it may hold for <math>z</math> from the reduced domain <math>\mathcal{D}</math>.
 
==Applications of superfunctions and the Abel functions==
<!--===Storage of huge numbers===!-->
 
Superfunctions, usially the [[tetration|superexponential]]s, are proposed as a fast-growing function for an
upgrade of the [[floating point]] representation of numbers in computers. Such an upgrade would greatly extend the
range of huge numbers which are still distinguishable from infinity.
 
Other applications refer to the calculation of fractional iterates
(or fractional power) of a function. Any holomorphic function can be declared as a "transfer function", then its superfunctions and
corresponding Abel functions can be considered.
===Transition from a function to its inverse function===
[[Image:Expc.jpg|140px|left|thumb|<math>\exp^c(x)</math> versus <math>x</math> for various <math>c</math>]]
[[Image:Sqrt(exp)(z).jpg|200px|right|thumb|<math>\exp^c</math> in the complex plane for various <math>c</math>]]
 
A superfunction <math>S</math> allows to calculate the fractional iteration <math>H^c</math> of some transfer function <math>H</math>. Once the superfunction <math>S</math> and the [[Abel function]] <math>A=S^{-1}</math> are established,
the fractional iteration can be defined as
<math>H^c(z)=S(c+A(z))</math>. Then, as <math>c</math> changes from 1 to <math>-1</math>, the holomorphic transition from function <math>H</math> to <math>H^{-1}</math> is relalised. The figure at left shows an example of transition from
<math>\exp^{1}\!=\!\exp </math> to
<math>\exp^{\!-1}\!=\!\ln </math>.
Function <math>\exp^c</math> versus real argument is plotted for
<math>c=2,1,0.9, 0.5, 0.1, -0.1,-0.5, -0.9, -1,-2</math>. The [[tetration]]al and ArcTetrational were used as superfunction
<math>S</math>
and Abel function <math>A</math> of the exponential.
The figure at right shows these functions in the complex plane.
At non-negative integer number of iteration, the iterated exponential is [[entire function]]; at non-integer values, it has two [[branch points]], thich correspond to the [[fixed points]] <math>L</math> and
<math>L^*</math> of natural logarithm. At <math>c\!\ge\! 0</math>, function <math>\exp^c(z)</math> remains [[holomorphic function|holomorphic]] at least in the strip <math>|\Im(z)|<\Im(L)\approx 1.3 </math> along the real axis.
 
===Nonlinear Optics===
In the investigation of the nonlinear response of optical materials,
the sample is supposed to be optically thin, in such a way,
that the intensity of the light does not change much as it goes through.
Then one can consider, for example, the absorption as function of the intensity.
However, at small variation of the intensity in the sample,
the precision of measurement of the absorption as function of intensity is not good.
The reconstruction of the superfunction from the Transfer Function allows to work with
relatively thick samples, improving the precision of measurements. In particular, the
Transfer Function of the similar sample, which is half thiner,
could be interpreted as the square root (id est, half-iteration) of the Transfer Function of the initial sample.
 
Similar example is suggested for a nonlinear optical fiber <ref name="kouznetsov">
{{cite journal
|author=D.Kouznetsov.
|title=Solutions of <math>F(z+1)=\exp(F(z))</math> in the complex <math>z</math>plane.
|journal=[[Mathematics of Computation]],
|year=2009
|volume=78
|pages=1647-1670
|url= http://www.ams.org/mcom/2009-78-267/S0025-5718-09-02188-7/home.html
|preprint: http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf
|doi=10.1090/S0025-5718-09-02188-7
}}</ref>.
 
===Nonlinear Acoustics===
It may have sense to characterize the nonlinearities in the
attenuation of shock waves in a homogeneous tube. This could find an application in some
advanced muffler, using nonlinear acoustic effects to withdraw the energy of the sound waves
without to disturb the flux of the gas. Again, the analysis of the nonlinear response,
id est, the Transfer Function,
may be boosted with the superfunction.
 
===Vaporization and condensation===
<!--For the separation of isotopes due to the different pressure of the saturated vapor for different components,!-->
In analysis of condensation, the growth (or vaporization) of a small drop of liquid can be considered,
as it diffuses down through a tube with some uniform concentration of vapor.
In the first approximation, at fixed concentration of the vapor,
the mass of the drop at the output end can be interpreted as the
Transfer Function of the input mass.
The square root of this Transfer Function will characterize the tube of half length.
 
===Snow avalanche===
The mass of a snowball, that rolls down from the hill,
can be considered as a function of the path it already have passed. At fixed length of this path
(that can be determined by the altitude of the hill) this mass can be considered also as a Transfer Function of the input mass. The mass of the snowball could be measured at the top of the hill and at thе bottom, giving the Transfer Function; then, the mass of the snowball as a function of the length it passed is superfunction.


===Operational element===
If one needs to build-up an operational element with some given transfer function <math>H</math>,
and wants to realize it as a sequential connection of a couple of identical operational elements, then, each of these two elements should have transfer function
<math> h=\sqrt{H}</math>. Such a function can be evaluated through the superfunction and the Abel function of the transfer function <math>H</math>.


==Applications of superfunctions and Abel functions==
The operational element may have any origin: it can be realized as an electronic microchip,
or a mechanical couple of curvilinear grains), or some asymmetric U-tube filled with different liquids, and so on.


==References==
==References==
<references/>
{{reflist}}[[Category:Suggestion Bot Tag]]

Latest revision as of 11:01, 23 October 2024

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Superfunction comes from iteration of another function. Roughly, for some function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} and for some constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} , the superfunction could be defined with expression

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {{S(z)} \atop \,} {= \atop \,} {{\underbrace{f\Big(f\big(... f(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}f\! \!\!\!\!\!}}}

then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} can be interpreted as superfunction of function . Such definition is valid only for positive integer Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} . The most research and applications around the superfunctions are related with various extensions of superfunction; and analysis of the existence, uniqueness and ways of the evaluation. For simple function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} , 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

logos of the Phys.Dept. of the MSU and that of Math.Dep.
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{!}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}} in the complex plane

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}~} ; then, function was used as logo of the Physics department of the Moscow State University [1][2][3]. (Mathematicians of the same University were not so arrogant and used the symbol of integral and the Moebius surface at their logo, see the figure at left).

That time, researchers did not have computational facilities for evaluation of such functions, but the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}} was more lucky than the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\sqrt{!~}~~} ; at least the existence of holomorphic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi} such that has been demonstrated in 1950 by Helmuth Kneser [4]. Actually, for his proof, Kneser had constructed the superfunction of exp and the corresponding Abel function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{X}} , satisfying the Abel equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{X}(\exp(z))=\mathcal{X}(z)+1} .

The inverse function, id est Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F=\mathcal \chi^{-1}} is an entire super-exponential, although it is not real at the real axis; it cannot be interpreted as tetrational, because the condition Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(0)=1} cannot be realized for the entire super-exponential. The real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}} can be constructed with the tetrational (which is also a superexponential), and the real Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\rm Factorial}} can be constructed with the superfactorial. The plots of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\rm Factorial}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{\exp}} in the complex plane are shown in the right hand side figure.

Extensions

The recurrent formula of the preamble can be written as equations

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(z\!+\!1)=f(S(z)) ~ \forall z\in \mathbb{N} : z>0}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(1)=f(t)} .

Instead of the last equation, one could write

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(0)=t}

and extend the range of definition of superfunction to the non-negative integers. Then, one may postulate

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(-1)=f^{-1}(t)}

and extend the range of validity to the integer values larger than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -2} . The following extension, for example,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(-2)=f^{-2}(t)}

is not trivial, because the inverse function may happen to be not defined for some values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} . In particular, tetration can be interpreted as super-function of exponential for some real base ; in this case,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f=\exp_{b}}

then, at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=1} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(-1)=\log_b(1)=0 } .

but

.

For extension to non-integer values of the argument, superfunction should be defined in different way.

Definition

For complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~a~} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~b~} , such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~a~} belongs to some connected domain ,
superfunction (from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} ) of holomorphic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~f~} on domain is function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S } , holomorphic on domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} , such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(z\!+\!1)=f(S(z)) ~ \forall z\in D : z\!+\!1 \in D}
.

Uniqueness

In general, the super-function is not unique. For a given base function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} , from given Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a\mapsto d)} superfunciton Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} , another Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a\mapsto d)} super-function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G} could be constructed as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle G(z)=F(z+\mu(z))}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu} is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mu(a)=0 } .

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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . In particular, the super-function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp_b} , for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>1} , is called tetration and is believed to be unique at least for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C= \{ z \in \mathbb{C} ~:~\Re(z)>-2 \}} ; for the case Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>\exp(1/\mathrm{e})} , see [6]; but up to year 2009, the uniqueness is rather conjecture than a theorem with the formal mathematical proof.

Examples

The short table of superfunctions is suggested in [7]. A little bit more extended table is available at TORI [8]. Some superfunctions can be expressed with elementary functions, they are used without to mention that they are superfunctions. For example, for the transfer function "++", which means unity increment, the superfunction is just addition of a constant.

Addition

Chose a complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} and define function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{add}_c} with relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{add}_c(z)=c\!+\!z ~ \forall z \in \mathbb{C}} . Define function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{mul_c}} with relation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{mul_c}(z)=c\!\cdot\! z ~ \forall z \in \mathbb{C}} .

Then, function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathrm{mul_c}~} is superfunction (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~0} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ c~} ) of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathrm{add_c}~} on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~\mathbb{C}~} .

Multiplication

Exponentiation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp_c} is superfunction (from 1 to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} ) of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{mul}_c } .

Quadratic polynomials

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(z)=2 z^2-1} . Then, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z)=\cos( \pi \cdot 2^z) } is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{C},~ 0\! \rightarrow\! 1)} superfunction of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} .

Indeed,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(z+1)=\cos(2 \pi \cdot 2^z)=2\cos(\pi \cdot 2^z)^2 -1 =H(f(z)) }

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0)=\cos(2\pi)=1}

In this case, the superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is periodic; its period

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T=\frac{2\pi}{\ln(2)} \mathrm{i}\approx 9.0647202836543876194 \!~i }

and the superfunction approaches unity also in the negative direction of the real axis,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\rightarrow -\infty} f(x)=1}

The example above and the two examples below are suggested at [9].

More general case refers to the logistic transfer function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(z)=u~z~(1-z) } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u>1} is parameter; the corresponding superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} is called logistic sequence [10]; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z\!+\!1)=H(F(z))} . Within wide range of values of parameter Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} , the logistic sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(z)} is holomorphic function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} ; at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u\!=\!4} , it can be expressed in a way, equivalent to the representation above [7] .

Rational function

In general, the transfer function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} has no need to be entire function. Here is the example with meromorphic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . Let

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(z)=\frac{2z}{1-z^2} ~ \forall z\in D~} ; Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ~ D=\mathbb{C} \backslash \{-1,1\}}

Then, function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z)=\tan(\pi 2^z)}

is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (C, 0\! \mapsto\! 0)} superfunction of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} is the set of complex numbers except singularities of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} . For the proof, the trigonometric formula

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tan(2 \alpha)=\frac{2 \tan(\alpha)}{1-\tan(\alpha)^2}~~ \forall \alpha \in \mathbb{C} \backslash \{\alpha\in \mathbb{C} : \cos(\alpha)=0 || \sin(\alpha)=\pm \cos(\alpha) \} }

can be used at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \alpha=\pi 2^z } , that gives

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(F(z))=\frac{2 \tan(\pi 2^z)}{1-\tan(\pi 2^z)} = \tan(2 \pi 2^z)=F(z+1) }

Algebraic function

In the similar way one can consider the transfer function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(z)=2z \sqrt{1-z^2}}

and

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z)=\sin(\pi 2^z)}

which is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (C,~ 0\!\rightarrow \!0)} superfunction of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C= \{z\in \mathbb C : \Re( \cos(\pi 2^z))>0 \}} .

Exponentiation

Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H(z)= \exp_b(z)} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C= \{ z \in \mathbb{C} : \Re(z)>-2 \}} . Then, tetration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{tet}_b } is a Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (C,~ 0\! \rightarrow\! 1)} superfunction of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp_b} .

Abel function

Inverse of superfunction can be interpreted as the Abel function.

For some domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E\subseteq \mathbb{C}} and some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u\in E} ,Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v\in \mathbb{C}} ,
Abel function (from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v } ) of function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F} with respect to superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} on domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E \in \mathbb{C}} is holomorphic function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S(A(z))=z ~\forall z \in E }
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(u)=c}

The definitionm above does not reuqire that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(S(z))=z ~\forall z \in D } ; although, from properties of holomorphic functions, there should exist some subset Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{D}\subseteq D} such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(S(z))=z ~\forall z \in \mathcal{D} } . In this subset, the Abel function satisfies the Abel equation.

Abel equation

The Abel equation is some equivalent of the recurrent equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(S(z))=S(z\!+\!1)}

in the definition of the superfunction. However, it may hold for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} from the reduced domain Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{D}} .

Applications of superfunctions and the Abel functions

Superfunctions, usially the superexponentials, are proposed as a fast-growing function for an upgrade of the floating point representation of numbers in computers. Such an upgrade would greatly extend the range of huge numbers which are still distinguishable from infinity.

Other applications refer to the calculation of fractional iterates (or fractional power) of a function. Any holomorphic function can be declared as a "transfer function", then its superfunctions and corresponding Abel functions can be considered.

Transition from a function to its inverse function

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp^c(x)} versus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} for various Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp^c} in the complex plane for various Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c}

A superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} allows to calculate the fractional iteration Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^c} of some transfer function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . Once the superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and the Abel function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A=S^{-1}} are established, the fractional iteration can be defined as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^c(z)=S(c+A(z))} . Then, as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} changes from 1 to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -1} , the holomorphic transition from function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^{-1}} is relalised. The figure at left shows an example of transition from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp^{1}\!=\!\exp } to . Function versus real argument is plotted for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c=2,1,0.9, 0.5, 0.1, -0.1,-0.5, -0.9, -1,-2} . The tetrational and ArcTetrational were used as superfunction Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} and Abel function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} of the exponential. The figure at right shows these functions in the complex plane. At non-negative integer number of iteration, the iterated exponential is entire function; at non-integer values, it has two branch points, thich correspond to the fixed points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^*} of natural logarithm. At Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c\!\ge\! 0} , function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exp^c(z)} remains holomorphic at least in the strip Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Im(z)|<\Im(L)\approx 1.3 } along the real axis.

Nonlinear Optics

In the investigation of the nonlinear response of optical materials, the sample is supposed to be optically thin, in such a way, that the intensity of the light does not change much as it goes through. Then one can consider, for example, the absorption as function of the intensity. However, at small variation of the intensity in the sample, the precision of measurement of the absorption as function of intensity is not good. The reconstruction of the superfunction from the Transfer Function allows to work with relatively thick samples, improving the precision of measurements. In particular, the Transfer Function of the similar sample, which is half thiner, could be interpreted as the square root (id est, half-iteration) of the Transfer Function of the initial sample.

Similar example is suggested for a nonlinear optical fiber [6].

Nonlinear Acoustics

It may have sense to characterize the nonlinearities in the attenuation of shock waves in a homogeneous tube. This could find an application in some advanced muffler, using nonlinear acoustic effects to withdraw the energy of the sound waves without to disturb the flux of the gas. Again, the analysis of the nonlinear response, id est, the Transfer Function, may be boosted with the superfunction.

Vaporization and condensation

In analysis of condensation, the growth (or vaporization) of a small drop of liquid can be considered, as it diffuses down through a tube with some uniform concentration of vapor. In the first approximation, at fixed concentration of the vapor, the mass of the drop at the output end can be interpreted as the Transfer Function of the input mass. The square root of this Transfer Function will characterize the tube of half length.

Snow avalanche

The mass of a snowball, that rolls down from the hill, can be considered as a function of the path it already have passed. At fixed length of this path (that can be determined by the altitude of the hill) this mass can be considered also as a Transfer Function of the input mass. The mass of the snowball could be measured at the top of the hill and at thе bottom, giving the Transfer Function; then, the mass of the snowball as a function of the length it passed is superfunction.

Operational element

If one needs to build-up an operational element with some given transfer function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} , and wants to realize it as a sequential connection of a couple of identical operational elements, then, each of these two elements should have transfer function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h=\sqrt{H}} . Such a function can be evaluated through the superfunction and the Abel function of the transfer function Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} .

The operational element may have any origin: it can be realized as an electronic microchip, or a mechanical couple of curvilinear grains), or some asymmetric U-tube filled with different liquids, and so on.

References

  1. ↑ Logo of the Physics Department of the Moscow State University. (In Russian); http://zhurnal.lib.ru/img/g/garik/dubinushka/index.shtml
  2. ↑ 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.

  3. ↑ 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.

  4. ↑ H.Kneser (1950). "Reelle analytische L¨osungen der Gleichung Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varphi(\varphi(x)) = e^x } und verwandter Funktionalgleichungen". Journal fur die reine und angewandte Mathematik 187: 56-67.
  5. ↑ P.Walker (1991). "Infinitely differentiable generalized logarithmic and exponential functions". Mathematics of computation 196: 723-733.
  6. ↑ 6.0 6.1 D.Kouznetsov. (2009). "Solutions of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle F(z+1)=\exp(F(z))} in the complex Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} plane.". Mathematics of Computation, 78: 1647-1670. DOI:10.1090/S0025-5718-09-02188-7. Research Blogging.
  7. ↑ 7.0 7.1 D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12. (Preprint ILS UEC, 2009: http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf )
  8. ↑ http://tori.ils.uec.ac.jp/TORI/index.php/Table_of_superfunctions
  9. ↑ Mueller. Problems in Mathematics. http://www.math.tu-berlin.de/~mueller/projects.html
  10. ↑ Kouznetsov, Dmitrii (2010). "Holomorphic extension of the logistic sequence". Moscow University Physics Bulletin (2): 91-98.
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