05/30/2008, 07:44 AM

galathaea Wrote:Code:`by the time he gets to chapter 4`

he is ready to return to iterated exponentiation

and after defining

F (x) = x

0

F (x) = exp{F (x)} - 1

r+1 r

he decomposes the iteration in two different ways

oo oo

--- ---

\ j \ j

F (x) = / phi (r) x = / f (x) r

r --- j --- j

j=0 j=0

I was really curious how Ramanujan's continuous iteration of would look like. But now I am a bit disappointed. What he considers is not iterated but iterated !

Not that this would be entirely trivial, however this is a case where the function to iterate has a fixed point at 0 and there is only one way to obtain (continuous/fractional/real/complex/analytic) iterates of the formal powerseries and that is regular iteration.

So, though its amazing that he considered the topic of regular iteration at such an early time, he does not contribute towards analytic tetration, where the difficulty is exactly this .