Function Repository Resource:

Tetration

Source Notebook

Compute the order-k tetration of a number

Contributed by: Siria Sadeddin

ResourceFunction["Tetration"][a,k]

computes the order-k tetration of the number a.

Details and Options

ResourceFunction["Tetration"][a,Infinity] gives a finite value if and only if a is in the convergence interval

ResourceFunction["Tetration"][a,Infinity] diverges when

k must be a non-negative integer or Infinity.

a must be a number different from 0.

Examples

Basic Examples (4)

Tetration of 2 with order 3:

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Symbolic tetration:

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Infinite tetration of a real number:

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Infinite tetration of ⅈ:

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Scope (3)

Compute infinite tetrations of real numbers inside the convergence interval:

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$Table[ResourceFunction["Tetration"][y, Infinity], {y, (1/E)^(E), E^(1/E), 0.1}]$

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Compute tetrations of real numbers within the divergence interval :

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$Table[ResourceFunction["Tetration"][y, 4], {y, 0 + $MinMachineNumber, (1/E)^(E), 0.01}]$

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Compute tetrations of real numbers within the divergence interval :

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$Table[ResourceFunction["Tetration"][y, 4], {y, E^(1/E), 3, 0.1}]$

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Properties and Relations (5)

Behavior of the tetration of real numbers inside the divergence interval :

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$GraphicsRow[ Table[ListLinePlot[ Table[ResourceFunction["Tetration"][x, n], {x, 0 + 0.0001, (1/E)^(E) - 0.0001, 0.0001}]], {n, {101, 200, 301, 400}}]]$

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Behavior of infinite tetration of real numbers inside the convergence interval :

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$GraphicsRow[ Table[ListLinePlot[ Table[ResourceFunction["Tetration"][x, n], {x, E^(\[Minus]E), E^(1/E), 0.01}]], {n, {2, 3, 4, 100, 1000, 10000}}]]$

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Evolution and stabilization of the tetration curve inside the convergence interval for increasing k in a single plot:

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$Table[Table[ ResourceFunction["Tetration"][x, n], {x, E^(\[Minus]E), E^(1/E), 0.01}], {n, 2, 100}] // ListLinePlot$

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Tetration convergence curve:

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$Table[ResourceFunction["Tetration"][x, Infinity], {x, E^(\[Minus]E), E^(1/E), 0.01}] // ListLinePlot$

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Behavior of the tetration of real numbers within the divergence interval :

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$GraphicsRow[ Table[ListLinePlot[ Table[ResourceFunction["Tetration"][x, n], {x, (E)^(1/E), 4, 0.1}]], {n, {1, 2, 3}}]]$

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The highly divergent property of tetration is easy to see from the previous plots.

Neat Examples (5)

Real part plot of tetration order 3 for a range of complex values:

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Imaginary part plot of tetration order 2:

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Norm plot of tetration order 3:

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Imaginary part of tetration order 2 of a number e^z, where z is a complex number:

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Real part of tetration order 2 of a number e^z, where z is a complex number:

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Possible Issues (3)

Tetration of a=0 is Indeterminate:

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Computing tetration for large values of k within the divergence interval may cause overflow, because the tetration exceeded the maximum machine number. You can obtain the maximum k for a given a as such:

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$MaxTetrationK[a_] := Module[{k, y}, k = 0; (**init k**) y = 1; (**init tetration**) Quiet[While[y != Overflow[], k = k + 1; y = ResourceFunction["Tetration"][N[a], k] ; ]]; Return[k - 1] ]$

Maximum tetration for a=12:

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If you want to avoid long execution times, use the function N to get an approximated result:

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Large values of k cause an overflow:

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Infinite tetration does not exists outside the convergence interval:

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Publisher

WolframSpecialProjects

Version History

1.0.0 – 22 October 2020

Source Metadata

Citation:
- Euler, L., "De formulis exponentialibus replicatus." Acta Sci. Petropol., vol. 1, 38–60, 1778.
- Galidakis, I. N., "The Problem of the Infinite Exponential." Ph.D. thesis, Agricultural University, Athens, Greece, 2015.

Related Resources

License Information

This work is licensed under a Creative Commons Attribution 4.0 International License

Wolfram Function Repository

Tetration

Details and Options