Editing Tetration (section)

==== Complex bases ====
[[File:Tetration period.png|thumbnail|alt=A colorful graph that shows the period getting much larger|Tetration by period]]
[[File:Tetration escape.png|thumbnail|alt=A colorful graph that shows the escape getting much larger|Tetration by escape]]

Since [[complex number]]s can be raised to powers, tetration can be applied to ''bases'' of the form {{math|''z'' {{=}} ''a'' + ''bi''}} (where {{mvar|a}} and {{mvar|b}} are real). For example, in {{math|{{sup|''n''}}''z''}} with {{math|''z'' {{=}} ''i''}}, tetration is achieved by using the [[principal branch]] of the [[natural logarithm]]; using [[Euler's formula]] we get the relation:

: <math>i^{a+bi} = e^{\frac{1}{2}{\pi i} (a + bi)} = e^{-\frac{1}{2}{\pi b}} \left(\cos{\frac{\pi a}{2}} + i \sin{\frac{\pi a}{2}}\right)</math>

This suggests a recursive definition for {{math|{{sup|''n''+1}}''i'' {{=}} ''a′'' + ''b′i''}} given any {{math|{{sup|''n''}}''i'' {{=}} ''a'' + ''bi''}}:
: <math>\begin{align}
  a' &= e^{-\frac{1}{2}{\pi b}} \cos{\frac{\pi a}{2}} \\[2pt]
  b' &= e^{-\frac{1}{2}{\pi b}} \sin{\frac{\pi a}{2}}
\end{align}</math>

The following approximate values can be derived:
{| class="wikitable"
|+Values of tetration of complex bases
|-
!scope="col"| <math display="inline">{}^{n}i</math>
!scope="col"| Approximate value
|-
!scope="row"| <math display="inline">{}^{1}i = i</math>
| {{math|''i''}}
|-
!scope="row"| <math display="inline">{}^{2}i = i^{\left({}^{1}i\right)}</math>
| {{math|0.2079}}
|-
!scope="row"| <math display="inline">{}^{3}i = i^{\left({}^{2}i\right)}</math>
| {{math|0.9472 + 0.3208''i''}}
|-
!scope="row"| <math display="inline">{}^{4}i = i^{\left({}^{3}i\right)}</math>
| {{math|0.0501 + 0.6021''i''}}
|-
!scope="row"| <math display="inline">{}^{5}i = i^{\left({}^{4}i\right)}</math>
| {{math|0.3872 + 0.0305''i''}}
|-
!scope="row"| <math display="inline">{}^{6}i = i^{\left({}^{5}i\right)}</math>
| {{math|0.7823 + 0.5446''i''}}
|-
!scope="row"| <math display="inline">{}^{7}i = i^{\left({}^{6}i\right)}</math>
| {{math|0.1426 + 0.4005''i''}}
|-
!scope="row"| <math display="inline">{}^{8}i = i^{\left({}^{7}i\right)}</math>
| {{math|0.5198 + 0.1184''i''}}
|-
!scope="row"| <math display="inline">{}^{9}i = i^{\left({}^{8}i\right)}</math>
| {{math|0.5686 + 0.6051''i''}}
|}

Solving the inverse relation, as in the previous section, yields the expected {{math|{{sup|0}}''i'' {{=}} 1}} and {{math|{{sup|−1}}''i'' {{=}} 0}}, with negative values of {{mvar|n}} giving infinite results on the imaginary axis.{{citation needed|date=January 2025}} Plotted in the [[complex plane]], the entire sequence spirals to the limit {{math|0.4383 + 0.3606''i''}}, which could be interpreted as the value where {{mvar|n}} is infinite.

Such tetration sequences have been studied since the time of Euler, but are poorly understood due to their chaotic behavior. Most published research historically has focused on the convergence of the infinitely iterated exponential function. Current research has greatly benefited by the advent of powerful computers with [[fractal]] and symbolic mathematics software. Much of what is known about tetration comes from general knowledge of complex dynamics and specific research of the exponential map.{{Citation needed|date=July 2019}}