Tetration

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Fig.1. Tetration for , , , and versus .

This article is currently under construction. While, use article from wikipedia http://en.wikipedia.org/wiki/Tetration

Tetration is fastly growing mathematical function, which was introduced in XX century and suggested for representation of huge numbers in mathematics of computation. However, up to year 2008, this function is not considered as elementary function, it is not implemented in programming languages and it is not used for the internal representation of data, at least in the commercial software.

Definiton

For real , Tetration on the base is function of complex variable, which is holomorphic at least in the range , bounded in the range , and satisfies conditions

at least within range .


Etymology

Creation of word tetration is attributed to Englidh mathematician Reuben Louis Goodstein [1] [2]. This name indicates, that this operation is fourth (id est, tetra) in the hierarchy of operations after summation, multiplication, and exponentiation. In principle, one can define "pentation", "sexation", "septation" in the simlar manner, although tetration, perhaps, already has growth fast enough for the requests of XXI century.

Real values of the arguments

Examples of behavior of this function at the real axis are shown in figure 1 for values , , , and for . It has logarithmic singularity at , and it is monotonously increasing function.

At tetration approaches its limiting value as , and .

At tetration grows faster than any exponential function. For this reason the tetration is suggested for the representation of huge numbers in mathematics of computation. A number, that cannot be stored as floating point, could be stored as for some standard value of (for example, or ) and relatively moderate value of . The analytic properties of tetration could be used for the implementation of arithmetic operations with huge numbers without to convert them to the floating point representation.

Integer values of the argument

For integer , tetration can be interpreted as iterated exponential:

and so on; then, the argument of tetration can be interpreted as number of exponentiations of unity. From definition it follows, that

and

Relation with Ackermann funcitons

At base , tetration is related to the Ackermann function:

where Ackermann function is defined for the positive integer values of its arguemtnrs with equations

Asymptotic behavior of tetration

Evaluation of tetration

As the asymptoric of tetration is crutually depend of base in ficinity of value , the evaluation procidure is different foe the cases , , and , and should be considered intependently.

Case

Case

Case

Existence and uniqueness of tetration

This section is not yet written. There is non-finished draft at User:Dmitrii Kouznetsov/Analytic Tetration.

Inverse of tetration

Iterated exponential and

Especially interesting, and in particular, for the qualitative breakthrough, id the case of iteration of natural exponential, id est, . Existence of the fractional iteration, and, in particular, existence of operation was demonstrated in 1950 by H.Kneser. [3]. However, that time, there was no computer facility for the evlauation of such an exotic function that ; perhaps, just absense of an apropriate plotter did not allow Kneser to plot the distribution of fractal exponential function in the complex plane for various values of .

See also

References

  1. "TETRATION, a term for repeated exponentiation, was introduced by Reuben Louis Goodstein". Earliest Known Uses of Some of the Words of Mathematics, http://members.aol.com/jeff570/t.html
  2. R.L.Goodstein (1947). "Transfinite ordinals in recursive number theory". Journal of Symbolic Logic 12.
  3. Kneser

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