A Notation Question (raising the highest value in pow-tower to a different power)

[jaydfox]

There is actually another notation you'll see some of use, when discussing iteration theory, Abel functions, etc.

\( \exp^{k}(x) \)

Here, k is taken to be the number of iterations of the exponential function.  I believe the mathematical term is "functional iteration".  This format is very easily confused with exponentiation.  For example, in trigonometry, you'll often see the sine and cosine functions raised to integer powers.  For example, the triple angle formula for the sine function is:

\( \sin{{3}{\theta}}={3}\sin{\theta} - {4}\sin^{3}{\theta} \)

Here, the sin^3 means to find the sine of theta, then raise that value to the third power, i.e., exponentiate:
\( \sin^{3}{\theta} = \(\sin{\theta}\)^{3} \)

However, with the functional iteration notation, k is taken to be the number of iterations:
\( \exp^{3}(x) = \exp\(\exp\(\exp(x)\)\) \)

This notation works very well for integer iterations, but it becomes poorly defined for non-integer values of k.  Negative values of k would imply iterations of the inverse of exponentiation, i.e., logarithms.

\( \exp^{-3}(x) = \log\(\log\(\log(x)\)\) \)