Table of superfunctions

For a given function $T$, called transfer function, the holomorphic solution $F$ of Transfer equation

$\!\!\!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ F(z+1)=T(F(z))$

is called superfunction with respect to $T$.

The inverse function, id est, $G=F^{-1}$ is called Abel function with respect to $T$; it satisfies the Abel equation

$\!\!\!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ G(T(z))=G(z)+1$

In any pair of holomorphic functions $F$, $G\!=\!F^{-1}$, function $F$ can be declared as superfunction, function $G$ can be declared as Abel function, and then the corresponding transfer function can be expressed as follows:

$\!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ T(z)=F(1+G(z))$

The table of examples of superfunctions can be constructed in a way, similar to the construction of a table of indefinite integrals. (Some special function is differentiated; if the result fits in one line of the table and cannot be easy expressed through other integrands already presented in table, then the result is qualified as new "integrand" and its integral is added to the table.) For superfunctions, any pair, function and its inverse function, are declared as superfunction and the Abel function; If the Transfer function by (3) can be simplified to fit the cell in the table, it is added to the Table of superfunctions. The example of table of superfunctions appeared in 2010 in the Moscow University Physics bulletin [1].

Table

$T(z)~$ $F(z)~$ $G(z)~$ Comments, keywords, refs
00 $~ c$ $~ c$ $~$ does not exist constant
01 $~z+1$ $~z+b$ $~z-b$ Unity increment, addition, substraction
02 $~z+b$ $~z\,b$ $~z/b$ $b\! \ne \! 0$, addition, multiplication, division
03 $~z\, b$ $~ b^z$ $~\log_b(z)$ $b\! \ne \! 0$, $b\ne 1 ~$ ; multiplication, exp, log
03a $~a+b\, z$ $\displaystyle ~ a \frac{1-b^z}{1-b}$ $\displaystyle ~\log_b\Big(1+\frac{b-1}{a}z \big)$ $a \ne 0 ~$, $~b\! \ne \! 0$, $b\ne 1 ~$ ; log
04 $~b^z$ $~\mathrm{tet}_b(z)$ $~\mathrm{ate}_b(z)$ $b\! > \! 1$, exp, tetration, arctetration
05 $~z^b$ $~\exp(b^z)$ $~\log_b(\ln(z))$ $b\ne \pm 1$ , Power function, log, exp
050 $~c z^{1+r}$ $~ c^{-1/r} \exp\big( (1\!+\!r)^z\, c^{1/r}\big)$ $~ \frac{ \ln\big(c^{-1/r}\, \ln(c^{1/r}\,z\,) \big)}{\ln(1+r)}$ $c\!>\!0$, $r\!>\!0$ , Power function
05a $\displaystyle \frac{-a^2}{z}$ $~\displaystyle a\, \tan\left(\frac{2}{\pi} z\right)$ $~\displaystyle \frac{2}{\pi} \arctan(z/a)$ Iterate of linear fraction
05b $\displaystyle \frac{-a^2}{z}$ $~\displaystyle a\, \tan\Big(\frac{2}{\pi} z\Big)$ $~\displaystyle \frac{2}{\pi} \arctan(z/a)$ Iterate of linear fraction
05c $\displaystyle \frac{z}{c+z}$ $~\displaystyle \frac{1-c}{1-c^z}$ $~\displaystyle \log_c \Big(1-\frac{1-c}{z}\Big)$ $c\ne 0$, $c\ne 1$; Iterate of linear fraction
05d $\displaystyle \frac{z}{1+z}$ $~1/z$ $~1/z$ $\displaystyle T^n(z)=\frac{z}{1+n z}$
06 $~\log_b(z)$ $~\mathrm{tet}_b(-z)$ $~-\mathrm{ate}_b(z)$ ArcTetration
07 $~ \ln(b+\mathrm{e}^z)$ $~ \ln(bz)$ $~\mathrm{e}^z/b$ Logarithm
08 $~(a^b+z^b)^{1/b}$ $~a z^{1/b}$ $~(z/a)^b$ Exponential
09 $~2 z^2-1$ $~\cos(2^z)$ $~\log_2(\arccos(z))$ cosinus, trigonometric functions
10 $~2 z^2-1$ $~ \cosh(2^z)$ $~\log_2(\mathrm{arccosh}(z))$ (compare to "09") Hyperbolic functions
11 $~ 2 z/(1\!-\!z^2)$ $~ \tan(2^z)$ $~ \log_2(\arctan(z))$ tangent
12 $~ 2z/(1\!+\!z^2)$ $~ \tanh(2^z)$ $~ \log_2\big( 2 \ln\Big( \frac{z+1}{z-1} \Big) \Big)$ Exponential
13 $~ z!$ $~ \mathrm{SuperFactorial}(z)$ $~ \mathrm{AbelFactorial}(z)$ Factorial, SuperFactorial, AbelFactorial [1]
14 $~ u\, z\, (1\!−\!z)$ $~ \mathrm{LogisticSequence}(z)$ $~ \mathrm{LogisticSequence}^{-1}(z)$ $u\!=\!\mathrm{const}$, Logistic sequence, [2]
15 $~ \mathrm{Doya}(z)$ $~ \mathrm{Tania}(z)$ $~ (z+\ln(z)−1)$ Doya, Tania, LambertW, WrightOmega [3]
16 $~ \mathrm{Keller}(z)$ $~ \mathrm{Shoka}(z)$ $~ \mathrm{ArcShoka}(z)$ Keller, Shoka, ArcShoka [3]
17 $\begin{array}{c} n/2~ , \mathrm{ ~~if~~ } n/2 \in \mathbb N \\\! \frac{3n\!+\!1)}{2} ,\mathrm{ ~if~ } \frac{n\!+\!1}{2} \in \mathbb N \end{array}$ $~ \mathrm{SubCollatz}(z)$ $~ \mathrm{ArcSubCollatz}(z)$ Collatz subsequence
18 $~\displaystyle \frac{a^2\!+\!2az\!+\!bz}{b-z}$ $~\displaystyle \frac{az\!+\!b}{1\!-\!z}$ $~\displaystyle \frac{z\!-\!b}{z\!+\!a}$ $~\displaystyle T^n(z)=\frac{a^2 n + (b\!+\!a\!+na)z}{a+b-an-nz}$
19 $~ \mathrm{zex}(z)\!=\! z \exp(z)$ $~ \mathrm{SuZex}(z)$ $~ \mathrm{AuZex}(z)$ Zex, LambertW, SuZex, AuZex
20 $~ \mathrm{tra}(z)\!=\! z\!+\! \exp(z)$ $\mathrm{SuTra}(z)\!=\!\ln\!\big(\mathrm{SuZex}(z)\big)\!$ $\mathrm{AuTra}(z)\!=\!\mathrm{AuZex}\! \big(\exp(z)\big)\!$ Trappmann function, SuTra, AuTra [4]
21 $~ \sin(z)$ $~ \mathrm{SuSin}(z)$ $~ \mathrm{AuSin}(z)$ sin, SuSin, AuSin
22 $\!\mathrm{Nem}_q(z)\!=\!z\!+\!z^3\!+\!qz^4\!$ $~ \mathrm{SuNem}_q(z)$ $~ \mathrm{AuNem}_q(z)$ Nemtsov function, SuNem, AuNem
$t(z)=P(T( Q(z)))$ $~ f(z)=P(F(z))$ $~ g(z)=G(Q(z))$ $P(Q(z))\!=\!z$, $T(z)=F(1\!+\!G(z))$

Extensions and uniqueness

The table above could be much longer. As it is mentioned, any pair of functions ($F$, $F^{-1}$) $=$ ($F$, $G$) can be interpreted as (superfunction, Abel function) for the transfer function $T=z\mapsto F(1+F^{-1}(z))$.

Also, the functions from the table can be combined: for a holomorphic functions $P$ and $Q=P^{-1}$ any line in the table (except the 0th) can be transformed as follows:

$F(z) \longrightarrow P(F(z))~$, $~G(z) \longrightarrow G(Q(z))~$, $~T(z) \longrightarrow P(T(Q(z)))$

In addition, the swap of the arguments is allowed:

$F \longrightarrow G~$, $~G \longrightarrow F~$, $~T(z) \longrightarrow G(1\!+\!F(z))$

The replacement is assumed to be performed in parallel, not sequentially.

The general methods of construction of superfunctions allow to implement the efficient algorithms for solution of the transfer equation (1) and the Abel equation (2) even for those transfer functions, that cannot be simply expressed through the special functions with equation (3).

In general, superfunctions are not unique. The new superfunction can be obtained by the translation of the argument; for example, the superfunction in the line "9" of the Table can be obtained from than in line "8" by the imaginary displacement of the argument. Also, the different fixed points of a transfer function can be used in the regular iteration method. Between the fixed points, the resulting superfunctions may agree with many decimal digits, but the deviation is easy to see for the complex values of the argument [5]. For uniqueness of the superfunction, the specification of its behavior in the complex plane is essential [6]. It seems, that the physically–meaningful superfunctions do not show the exponentical growth in the imaginary direction and can be constructed at the real fixed point with the regular iterations.

From the scientific point of view, it is difficult to say, what is "better" – to have the variety of superfunctions for each case of the life needs, or to have the unique superfunction, applying the strict requirements on its behavior at the complex values of the argument. But anyway, it is better to know about this variety, while it takes place, and to know about the uniqueness in the cases, when the superfunction and the Abel function are unigue [7].

Referenes

1. http://www.ils.uec.ac.jp/~dima/PAPERS/2009supefae.pdf D.Kouznetsov, H.Trappmann. Superfunctions and square root of factorial. Moscow University Physics Bulletin, 2010, v.65, No.1, p.6-12.
2. http://www.springerlink.com/content/u712vtp4122544x4/ D.Kouznetsov. Holomorphic extension of the logistic sequence. Moscow University Physics Bulletin, 2010, No.2, p.91-98.
3. http://link.springer.com/article/10.1007/s10043-013-0058-6 (official version, registration is required)