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A Tetration is defined as $${^{n}a} = \underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_n$$ or, by a recursion function, $${^{n}a} := \begin{cases} 1 &\text{if }n=0 \\ a^{\left[^{(n-1)}a\right]} &\text{if }n>0 \end{cases} $$ That is, iterated exponentiation (like exponentiation is iterated multiplication). Exponentiation can be expressed as $e^{n\ln a}$ [1]

Is there a like way to express Tetrations? That is, a way to calculate for both negative and decimal Tetrations?

Conor O'Brien
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  • The comments at [Arrow notation and decimals.](http://math.stackexchange.com/questions/546032/arrow-notation-and-decimals) may be helpful. – MJD Nov 30 '14 at 23:32
  • Also relevant: [Is there a natural way to extend repeated exponentiation beyond integers?](http://math.stackexchange.com/questions/56663/is-there-a-natural-way-to-extend-repeated-exponentiation-beyond-integers/56710#56710) – MJD Nov 30 '14 at 23:34

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