(09/17/2009, 10:52 AM)mike3 Wrote: while \( f^n(x) \) is iteration (except for trig functions with positive (and integer?) superscript, in which case it refers to exponentiation of the function, apparently due to historical reasons.).
Not really for historical reasons. Whenever you write \( X^n \) you mean \( X \) multiplied \( n \) times. However it is not always clear what the multiplication is.
For sets it can perhaps be the cross product, or elementwise multiplication. For functions it can be composition or multiplication. As multiplication is more common than composition, I always would mark a compositional power differently, for example \( f^{\circ n} \) (which is used by Ecalle and in modern texts about complex dynamics) or \( f^{[n]} \) or \( f^{\langle x\rangle} \) (which is imho rather used in mathematical physics and general dynamic systems).
And it seems that Gottfried took the mark \( f^{(n)} \) as compositional power, perhaps because it is similarly round like \( f^{\circ n} \).
PS: though this discussion is a bit off this thread, there are notation threads to discuss those topics.