Tetration: Difference between revisions

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At <math>b > \exp(1/\mathrm{e})</math> tetration <math>\mathrm{tet}_b(x)</math> grows faster than any exponential function. For this reason the tetration is suggested for the representation of huge numbers in [[mathematics of computation]].
At <math>b > \exp(1/\mathrm{e})</math> tetration <math>\mathrm{tet}_b(x)</math> 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 <math>\mathrm{tet}_b(x)</math> for some standard value of <math>b</math> (for example, <math>b=2</math> or <math>b=\mathrm{e}</math>) and relatively moderate value of <math>x</math>. The analytic properties of tetration could be used for the implementation of arithmetic operations without to convert numbers to the floating point representation.
A number, that cannot be stored as [[floating point]], could be stored as <math>\mathrm{tet}_b(x)</math> for some standard value of <math>b</math> (for example, <math>b=2</math> or <math>b=\mathrm{e}</math>) and relatively moderate value of <math>x</math>. 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==
==Integer values of the argument==

Revision as of 06:39, 29 October 2008

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

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

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 .

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

Etymology

Creation of word tetration is attributed to Englidh mathematician Reuben Louis Goodstein [1] [2].

Piecewice tetration

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Analytic tetration

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

Inverse of tetration

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.

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