Tetration: Difference between revisions

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<math>x</math>.]]
<math>x</math>.]]
This article is currently [[under construction]]. While, use article from wikipedia http://en.wikipedia.org/wiki/Tetration
This article is currently [[under construction]]. While, use article from wikipedia http://en.wikipedia.org/wiki/Tetration
==definiton==
==Definiton==
For real <math>b>1</math>, Tetration <math>F=\mathrm{tet}_b</math> on the base <math>b</math> is function of complex variable, which is [[holomorphic]] at least in the range
<math> \{ z \in \mathbb{C} :~ \Re(z) > -2 \}</math>, bounded in the range
<math> \{ z \in \mathbb{C} :~ |\Re(z)| \le 1 \}</math>, and satisfies conditions
: <math>  F(z+1) = \exp_b\! \big( F(z) \big) </math>
: <math>  F(0) = 1 </math>
: <math>  F\!\big(z^*\big) = F\big(z\big)^*</math>
at least within range <math> \Re(z)>-2 </math>.


==Real values of the arguments==
Examples of behavior of this function at the real axis are shown in figure 1 for values
<math>b=\mathrm{e}</math>,
<math>b=2</math>,
<math>b=\exp(1/\mathrm{e})</math>, and for
<math>b=\sqrt{2}</math>. It has logarithmic singularity at <math>-2</math>, and it is monotonously increasing function.
At <math>b\le \exp(1/\mathrm{e})</math> tetration <math>\mathrm{tet}_b(x)</math> approaches its limiting value as
<math>x\rightarrow +\infty</math>, and <math>\displaystyle  \lim_{x \rightarrow +\infty}  ~ \mathrm{tet}_b(x) > 1</math>.
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.
==Integer values of the argument==
For integer  <math>z</math>, tetration
: <math>{\rm tet}_b</math>


==Etymology==
==Etymology==

Revision as of 06:34, 29 October 2008

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 without to convert numbers 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

uxp

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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