05/27/2008, 08:12 AM

Hi, just today I found this msg in the sci.math newsgroup, which may be of interest here. Especially the second function of Ramanujan, which combines (a somehow inverse to) Andrew's E()-function and the ("Tetra-")series of increasing heigths. Maybe, you like this

Gottfried

subject: yeah sure!!

author: galathaea@gmail.com

Gottfried

subject: yeah sure!!

author: galathaea@gmail.com

Code:

`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

(...)

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galathaea: prankster, fablist, magician, liar

Gottfried Helms, Kassel