Introduction.
I begin with this introduction in order to indicate, how do I understand the super-functions and our role about them. I remember that the Moderator has an ability to remove this introduction, together with all the lyrics around (and I appreciate his good will to keep all the posts so dry as his gunpowder); however, I hope the definitions below do not contradict those he suggested for our joint paper (which is "yet to be finished" during several last months); so, some definitions have some chance to survive.
Terminology
'''Superfunction''' \( S \) comes from iteration of some given function \( h \), called "base-function" or "transfer function". There is some analogy with fiber optics, which explains why this \( h \) should be called "transfer function". Those who hate any physics (and, especially, the phenomenological fiber optics), may imagine that the function \( h \) transfers the value of function \( S \) at some point \( z \) to the value at the point \( z+1 \), as the basic equation suggests:
\( S(z+1)=f(S(z)) \)
This equation is very basic; so, the only given function \( h \) may also be called the "base-function".
Iterations
Roughly, for some function \( h \) and for some constant \( t \), the super-function could be defined with expression
\( {{S(z)} \atop \,} {= \atop \,}
{{\underbrace{h\Big(h\big(... h(t)...\big)\Big)}} \atop {z \mathrm{~evaluations~of~function~}h\!\!\!\!\!\!}} \)
then \( S \) can be interpreted as superfunction of function \( h \).
Such definition is valid only for positive integer \( z \).
In particular, \( S(1)=h(t) \).
The most research and applications around the superfunctions are related with various extensions of super-function: analysis of the existence, uniqueness and ways of the evaluation.
For some functions \( h \), such as addition of a constant or multiplication by a constant,
the superfunction can be expressed in terms of elementary function.
Namely such examples were motivation of this message.
History and Lowstory
Analysis of superfunctions cames from the application to the evaluation of fractional iterations of functions. Super-functions 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 \( \sqrt{\exp}~ \); then, function \( \sqrt{!~}~ \) was used as logo of the Physics department of the Moscow State University, see
http://zhurnal.lib.ru/img/g/garik/dubinu...ndex.shtml
http://ofvp.phys.msu.ru/pdf/Kandidov_70.pdf
http://nauka.relis.ru/11/0412/11412002.htm
(bitte, all 3 in Russian).
That time, researchers did not have computational facilities for evaluation of such functions, but
the \( \sqrt{\exp} \) was more lucky than the \( ~\sqrt{!~}~~ \); at least the existence of <b>holomorphic function</b>
\( \varphi \) such that \( \varphi(\varphi(z))=\exp(z) \) has been reported in 1950 by <b>Helmuth Kneser</b>
(H.Kneser. “Reelle analytische L¨osungen der Gleichung \( \varphi(\varphi(x)) = e^x \) und verwandter Funktionalgleichungen”.
Journal fur die reine und angewandte Mathematik, 187 (1950), 56-67.)
Extensions
The recurrence above can be written as equations
\( S(z\!+\!1)=h(S(z)) ~ \forall z\in \mathbb{N} : z>0 \)
\( S(1)=h(t) \).
Instead of the last equation, one could write
\( S(0)=t \)
and extend the range of definition of superfunction \( S \) to the non-negative integers. Then, one may postulate
\( S(-1)=h^{-1}(t) \)
and extend the range of validity to the integer values larger than \( -2 \).
The following extension, for example,
\( S(-2)=h^{-2}(t) \)
is not trivial, because the inverse function may happen to be not defined for some values of \( t \).
In particular, [[tetration]] can be interpreted as super-function of exponential for some real base \( b \); in this case,
\( h=\exp_{b} \)
then, at \( t=1 \),
\( S(-1)=\log_b(1)=0 \).
but
\( S(-2)=\log_b(0)~ \mathrm{is~ not~ defined} \).
For extension to non-integer values of the argument, superfunction should be defined in different way.
Definitions.
For connected domains \( C \subseteq \mathbb{C} \) and \( D \subseteq \mathbb{C}, and two numbers [tex]a\in C \) and \( d\in D \),
the \( (a \!\mapsto\! d) \) super-function of a transfer function \( ~h~ \)
is function \( S \), holomorphic on \( C \), such that
\( S(z\!+\!1)=h(S(z)) ~ \forall z\in C : S(z)\in D \) and \( S(a)=b \).
If \( h=\exp_b [tex] for some [tex]b\in \mathbb{C} \),
then the \( (a \!\mapsto\! d) \) super-function of a transfer function \( ~h~ \) is called \( (a \!\mapsto\! d) \) super-exponential on the base \( b \).
If \( a=0 \) and \( d=1 \),
then such a super-exponential is called <b>tetration</b>
and justify the appearance of this post at this Forum.
As it was already mentioned in this forum, in general, the super-function is not unique.
For a given transfer function \( h \), from given \( (a\mapsto d) \) super-funciton \( F \), another \( (a\mapsto d) \) super-function \( G \) could be constructed as
\( G(z)=F(z+\mu(z)) \)
where \( \mu \) is any 1-periodic function, holomorphic at least in some vicinity of the real axis, such that \( \mu(a)=0 \).
The modified super-function may have narrowed range of holomorphism.
The challenging task is to specify some domain \( C \) such that
\( (C, a \mapsto d) \) super-function is unique.
In particular, the \( (C, 0\mapsto 1) \) super-function of
\( \exp_b \), for \( b>1 \), is called [[tetration]] and is believed to be unique at least for
\( C= \{ z \in \mathbb{C} ~:~\Re(z)>-2 \} \); for the case \( b>\exp(1/e) \)
Examples
Oh, en fin, I touch the goal of this post. Sorry for the long introduction above.
Below, I consider various simple base-functions \( h \) .
<b>Elementary increment</b> Let \( h(z)=z+1=++z \).
Then, the identity function \( I \) such that \( I(z)=z \forall z \in \mathbb{C} \)
is \( (\mathbb{C}, 0\mapsto 0) \) superfunction of \( h \).
Addition
Chose a \( b \) and define function \( \mathrm{add}_b \) such that
\( \mathrm{add}_b(z)=b\!+\!z ~ \forall z \in \mathbb{C} \)
Define function \( \mathrm{mul_b} \) such that
\( \mathrm{mul_b}(z)=b\!\cdot\! z ~ \forall z \in \mathbb{C} \).
Then, function \( ~\mathrm{mul_b}~ \) is \( (\mathbb{C}, 0 \mapsto b ) \)
superfunction of \( h \).
<b>Multiplication</b>
Exponential \( \exp_b \) is \( (\mathbb{C}, 0 \mapsto 1 ) \)
super-function of function \( \mathrm{mul}_b \), defined in the previous example.
Quadratic polynomial
Let \( h(z)=2 z^2-1 \).
Those, who like some Quantum Mechanics, may treat this function as a scaled second Hermitian polynomial, justifying the letter, used to denote the transfer function.
Then, \( f(z)=\cos( \pi \cdot 2^z) \) is a
\( (\mathbb{C},~ 0\! \rightarrow\! 1) \) superfunction of \( H \).
Indeed, \( f(z+1)=\cos(2 \pi \cdot 2^z)=2\cos(\pi \cdot 2^z)^2 -1 =H(f(z)) \)
and \( f(0)=\cos(2\pi)=1 \)
In this case, the superfunction \( f \) is periodic; its period
\( T=\frac{2\pi \mathrm {i}}{\ln(2)} \mathrm{i}\approx 9.0647202836543876194 \!~i \).
Such super-function approaches unity in the negative direction of the real axis,
\( \lim_{x\rightarrow -\infty} f(x)=1 \)
The example above and the two examples below are suggested at
<ref name="mueller">Mueller. Problems in Mathematics.
http://www.math.tu-berlin.de/~mueller/projects.html
</ref>
Rational function. In general, the transfer function \( h \) has no need to be entire function. Here is the example with meromorphic function \( h \).
Let
\( C= \{z \in \mathbb{C}: 2^z\ne n+1/2 \forall n\in \mathbb{Z})\} \)
\( D= \mathbb{C}\backslash \{1,-1\} \)
\( h(z)=\frac{2z}{1-z^2} ~ \forall z\in D~ \)
\( S(z)=\tan(\pi 2^z) \forall z\in C \)
Tthen, \( S \) is \( (C, 0\! \mapsto\! 0) \) superfunction of \( h \).
For the proof, the trigonometric formula
\( \tan(2 \alpha)=\frac{2 \tan(\alpha)}{1-\tan(\alpha)^2}~~
\forall \alpha \in \mathbb{C} \backslash \{\alpha\in \mathbb{C} : \cos(\alpha)=0 || \sin(\alpha)=\pm \cos(\alpha) \}
\)
can be used at \( \alpha=\pi 2^z \), that gives \(
h(S(z))=\frac{2 \tan(\pi 2^z)}{1-\tan(\pi 2^z)} = \tan(2 \pi 2^z)=S(z+1) \)
Algebraic transfer function. However, the transfer function has no need to be even meromorphic. Let
\( C= \{z \in \mathbb{C}: |Arg(cos(\pi 2^z))| < \pi \} \)
\( D= \{z \in \mathbb{C}: |Arg(1-z^2)| < \pi \} \)
\( h(z)=2z \sqrt{1-z^2} \forall z \in D \)
\( S(z)=\sin(\pi 2^z) \forall z \in C \)
Then, \( S is is [tex](C,~ 0\!\rightarrow \!0) \) superfunction of \( H \) for
\( C= \{z\in \mathbb C : \Re( \cos(\pi 2^z))>0 \} \).
The proof is similar to the previous two cases.
Exponential transfer function. Let \( b>1 \),
\( H(z)= \exp_b(z) \forall z \in \mathbb{C} \),
\( C= \{ z \in \mathbb{C} : \Re(z)>-2 \} \).
Then, tetrational \( \mathrm{tet}_b \)
is a \( (C,~ 0\! \rightarrow\! 1) \) super-function of \( \exp_b \).
more extensions.
In general, we may take any special function \( S \), such that \( S(z+1) \) can be expressed through \( S(z) \) with holomorphic elementary functions, then we may declare this expression as transfer function \( h \), and then, function \( S \) appears as super-function. I invite participants to construct more super-functions that can be easy represented through some already known special functions.
P.S. Oh, mein Gott! I just realized the correct tread for this post. It repeats a lot of staff already posted here... Sorry... I see, there are already replies, so, I ssto to edit; the only correct obvious misprints...
