Iteration by Ramanujan

[bo198214]

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i've already pointed out in this thread
that ramanujan worked on continuous iteration in his quarterly reports

these were written between august 5th, 1913
  and march 9th, 1914

ramanujan actually expands the notion of iteration
  into a power series

            oo            j
           ---  psi (x)  n
           \       j
(f)^n(x) = /    -----------
           ---      (1)
           j=0         j

where
  because n could be any value in the convergence radius
there is a potential continuous definition

but ramanujan was by no means the first either

i've also mentioned comtet's book
  which even berndt's coverage recommends
to put this on a rigorous foundation

this has been around since before euler

fractional differentiation
  for instance
was developed from several different transform approaches
from the very early transform studies

it's natural that if
  (-ik)^n corresponds to n-th differentiation
  in the transform language
  then there is a clear generalisation of differentiation
    that allows real orders

..

to show some of the other cool things in ramanujan's reports
and to connect to the tetration threads
  ramanujan studies

                         x
                  x     e
            x    e     e
           e    e     e
f(x) = 1 + -- + --- + ---- + ...
            3     4      5
           2     3      4
                2      3
                      2

ramanujan shows that this function is enitre
and yet grows faster than any
      x
     .
    .
   e
  e
finitely iterated exponential

Wow thats amazing, can you give any references?

@Gottfried: can you point galthaea to our forum?