Super-logarithm

Super-logarithm

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In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions: roots and logarithms, likewise tetration has two inverse functions: super-roots and super-logarithms. There are several ways of interpreting super-logarithms:

As the Abel function of exponential functions,
As the inverse function of tetration with respect to the height,
As the number of times a logarithm must be iterated to get to 1,
As the continuous extension of the iterated logarithm to the real numbers (see above),
As a generalization of Robert Munafo's large number class system,
As an unbounded version of the logistic function, or
As a function with a pathologically slow rate of growth.

One of the reasons why the super-logarithm has been slow to become standard is that there are several competing definitions. Each of these definitions complement each other in some way, and are all "correct" in some sense. So determining which definition is more "correct" has led to some confusion and debate amongst the mathematical community investigating the super-logarithm.

1 Definitions
2 Approximations
- 2.1 Linear approximation
- 2.2 Quadratic approximation
3 Approaches to the Abel function
- 3.1 Regular approach
- 3.2 Other approaches
4 Properties
5 slog as inverse of tetration
6 See also
7 References
8 External links

Definitions

The super-logarithm, written $\,\mathrm{slog}_b(z)$ , is defined implicitly by

$\,\mathrm{slog}_b(b^z) = \mathrm{slog}_b(z) + 1$ and

$\,\mathrm{slog}_b(1) = 0.$

Notice that this definition can only have integer outputs, and will only accept values that will produce integer outputs. The only numbers that this definition will accept are of the form $b, b^b, b^{b^b}$ and so on. In order to extend the domain of the super-logarithm from this sparse set to the real numbers, several approaches have been pursued. These usually include a third requirement in addition to those listed above, which vary from author to author. These approaches are:

The linear approximation approach by Rubstov and Romerio,
The quadratic approximation approach by Andrew Robbins,
The regular Abel function approach by George Szekeres,
The iterative functional approach by Peter Walker, and
The natural matrix approach by Peter Walker, and later generalized by Andrew Robbins.

Approximations

Usually, the special functions are defined not only for the real values of argument(s), but to complex plane, and differential and/or integral representation, as well as expansions in convergent and asymptotic series. Yet, no such representations are available for the slog function. Nevertheless, the simple approximations below are suggested.

Linear approximation

The linear approximation to the super-logarithm is:

$\mathrm{slog}_b(z) \approx \begin{cases} \mathrm{slog}_b(b^z) - 1 & \text{if } z \le 0 \\ -1 + z & \text{if } 0 < z \le 1 \\ \mathrm{slog}_b(\log_b(z)) + 1 & \text{if } 1 < z \\ \end{cases}$

which is a piecewise-defined function with a linear "critical piece". This function has the property that it is continuous for all real z ( $C 0$ continuous). The first authors to recognize this approximation were Rubstov and Romerio, although it is not in their paper, it can be found in their algorithm that is used in their software prototype. The linear approximation to tetration, on the other hand, had been known before, for example by Ioannis Galidakis. This is a natural inverse of the linear approximation to tetration.

Authors like Holmes recognize that the super-logarithm would be a great use to the next evolution of computer floating-point arithmetic, but for this purpose, the function need not be infinitely differentiable. Thus, for the purpose of representing large numbers, the linear approximation approach provides enough continuity ( $C 0$ continuity) to ensure that all real numbers can be represented on a super-logarithmic scale.

Quadratic approximation

The quadratic approximation to the super-logarithm is:

$\mathrm{slog}_b(z) \approx \begin{cases} \mathrm{slog}_b(b^z) - 1 & \text{if } z \le 0 \\ -1 + \frac{2\log(b)}{1+\log(b)}z + \frac{1-\log(b)}{1+\log(b)}z^2 & \text{if } 0 < z \le 1 \\ \mathrm{slog}_b(\log_b(z)) + 1 & \text{if } 1 < z \end{cases}$

which is a piecewise-defined function with a quadratic "critical piece". This function has the property that it is continuous and differentiable for all real z ( $C 1$ continuous). The first author to publish this approximation was Andrew Robbins in this paper.

This version of the super-logarithm allows for basic calculus operations to be performed on the super-logarithm, without requiring a large amount of solving beforehand. Using this method, basic investigation of the properties of the super-logarithm and tetration can be performed with a small amount of computational overhead.

Approaches to the Abel function

Main article: Abel function

The Abel function is any function that satisfies Abel's functional equation:

$\,A_f(f(x)) = A_f(x) + 1$

Given an Abel function $A f (x)$ another solution can be obtained by adding any constant $A' f (x) = A f (x) + c$ . Thus given that the super-logarithm is defined by $slog b (1) = 0$ and the third special property that differs between approaches, the Abel function of the exponential function could be uniquely determined.

Regular approach

Szekeres defines the regular Abel function as the logarithm of the regular Schroeder function. The regular Schroeder function is defined as

$S_f(x) = \lim_{n\rightarrow\infty} \frac{f^n(x)}{(f'(x_0))^n}$

(where $f n (x)$ is functional iteration, and $x 0 = f (x 0)$ is a fixed point of $f (x)$ ) so as to satisfy the functional equation (called Schroeder's functional equation)

$\,S_f(f(x)) = (f'(x_0))S_f(x)$

So using this, the regular Abel function is defined by

$A_f(x) = \frac{\log(S_f(x))}{\log(f'(x_0))}$

so as to satisfy the functional equation (called Abel's functional equation)

$\,A_f(f(x)) = A_f(x) + 1$

Using this approach, the super-logarithm is defined as $\mathrm{slog}_b(z) = A_{(\exp_b)}(x)$ , the regular Abel function of $b x$ . This is one of the oldest approaches, and as such it is called the regular approach. So the super-logarithm based on this approach is called the regular super-logarithm. Since the regular approach depends upon a fixed point $x 0$ , this is also called the regular super-logarithm about $x 0$ , for clarity.

Other approaches

Other approaches to the super-logarithm do exist, as mentioned above. For example, Peter Walker's approaches, both his iterative approach and his matrix approach are apt at discovering more about the super-logarithm. Until more papers are published on these methods, many aspects of these techniques remain unknown. For more information about these methods, the reader is encouraged to see the references below.

Properties

Other equations that the super-logarithm satisfies are:

$\,\mathrm{slog}_b(z) = \mathrm{slog}_b(\log_b(z)) + 1$

$\,\mathrm{slog}_b(z) > -2$ for all real z

slog as inverse of tetration

f = slog e (z)

in the complex z-plane.

As tetration (or super-exponential) $~{\rm sexp}_b(z)~$ is suspected to be an analytic function ¹, at least for some values of $~b~$ , the inverse function slog_b=sexp_b^-1 may also be analytic. Behavior of $~{\rm slog}_b(z)~$ , defined in such a way, the complex $~z~$ plane is sketched in Figure 1 for the case $~b=e~$ . Levels of integer values of real and integer values of imaginary parts of the slog functions are shown with thick lines. If the existence and uniqueness of the analytic extension of tetration is provided by the condition of its asymptotic approach to the fixed points $L \approx 0.318131505204764135312654 + 1.33723570143068940890116{\!~\rm i}$ and $L^* \approx 0.318131505204764135312654 - 1.33723570143068940890116{\!~\rm i}$ of $L = ln(L)$ ² in the upper and lower parts of the complex plane, then the inverse function should also be unique. Such a function is real at the real axis. It has two branch points at $~z=L~$ and $~z=L^*$ . It approaches its limiting value $- 2$ in vicinity of the negative part of the real axis (all the strip between the cuts shown with pink lines in the figure), and slowly grows up along the positive direction of the real axis. As the derivative at the real axis is positive, the imaginary part of slog remains positive just above the real axis and negative just below the real axis. The existence, uniqueness and generalizations are under discussion ³.

References

^ Peter Walker (1991). "Infinitely Differentiable Generalized Logarithmic and Exponential Functions". Mathematics of Computation, 57 (196): 723-733, http://www.jstor.org/pss/2938713.
^ H.Kneser (1950). "Reelle analytische Losungen der Gleichung $\varphi\Big(\varphi(x)\Big)={\rm e}^x$ und verwandter Funktionalgleichungen". Journal für die reine und angewandte Mathematik 187: 56-67.
^ Tetration forum, http://math.eretrandre.org/tetrationforum/index.php

Ioannis Galidakis, Mathematics, published online (accessed Nov 2007).
W. Neville Holmes, Composite Arithmetic: Proposal for a New Standard, IEEE Computer Society Press, vol. 30, no. 3, pp. 65-73, 1997.
Robert Munafo, Large Numbers at MROB, published online (accessed Nov 2007).
C. A. Rubtsov and G. F. Romerio, Ackermann's Function and New Arithmetical Operation, published online (accessed Nov 2007).
Andrew Robbins, Solving for the Analytic Piecewise Extension of Tetration and the Super-logarithm, published online (accessed Nov 2007).
George Szekeres, Abel's equation and regular growth: variations on a theme by Abel, Experiment. Math. Volume 7, Issue 2 (1998), 85-100.
Peter Walker, Infinitely Differentiable Generalized Logarithmic and Exponential Functions, Mathematics of Computation, Vol. 57, No. 196 (Oct., 1991), pp. 723-733.

External links

Rubstov and Romerio, Hyper-operations Thread 1
Rubstov and Romerio, Hyper-operations Thread 2

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