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 Iteration by Ramanujan
05/27/2008, 08:12 AM
Post: #1
 Gottfried Long Time Fellow Posts: 731 Joined: Aug 2007
Iteration by Ramanujan
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

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 (...) -=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=- galathaea: prankster, fablist, magician, liar```

Gottfried Helms, Kassel
05/27/2008, 08:23 AM
Post: #2
 bo198214 Administrator Posts: 1,366 Joined: Aug 2007
RE: Iteration by Ramanujan
galthaea Wrote:
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```

Wow thats amazing, can you give any references?

@Gottfried: can you point galthaea to our forum?
05/27/2008, 09:39 AM
Post: #3
 Gottfried Long Time Fellow Posts: 731 Joined: Aug 2007
RE: Iteration by Ramanujan
bo198214 Wrote:@Gottfried: can you point galthaea to our forum?
Henryk - I'll try to send him a personal mail; hope his adress is valid.

Gottfried

Gottfried Helms, Kassel
05/27/2008, 11:03 AM (This post was last modified: 05/27/2008 03:37 PM by Ivars.)
Post: #4
 Ivars Long Time Fellow Posts: 366 Joined: Oct 2007
RE: Iteration by Ramanujan
Are those 2^3,2^3^4, 2^3^4^5... = 2^3^4^...n what Andrew calls E factorial as E(n) ? Does it have generalization to x? Obviously they can not start at 1. The smallest integer is 2, unlike ordinary factorial.

What if its turned around, so 2^1, 3^2^1, 4^3^2^1, 5^4^3^2^1.. n^..3^2^1. It is also a fast growing number. x^(x-1)^(x-2)..1, but much slower then the other.

This slower one has been called exponential factorial:

Exponential Factorial Wolfram MathWorld

It is given by recurence relation:

$a_n=n^{a_{n-1}}$
$a_1=1$

Ramanujan's factorial would be bigger.

Do I understand right that by applying some transformation involving such factorials the summation of many divergent series can be brought to some sort of convergence-if their speed of growth is slower than these factorials?

Then these perhaps can be applied to power series of extremely slow functions directly, like e.g. 1/h(z).

Ivars
05/27/2008, 06:23 PM
Post: #5
 andydude Long Time Fellow Posts: 505 Joined: Aug 2007
RE: Iteration by Ramanujan
First of all, that is not the exponential factorial. The EF is (5^4^3^2) whereas this is (2^3^4^5). Secondly, I found this on JSTOR, so I'm going to make a trip to the local library soon...

Andrew Robbins
05/27/2008, 08:38 PM
Post: #6
 Ivars Long Time Fellow Posts: 366 Joined: Oct 2007
RE: Iteration by Ramanujan
andydude Wrote:Secondly, I found this on JSTOR, so I'm going to make a trip to the local library soon...

Andrew Robbins

One place Ramanujan considers infinite exponentials is Notebook 5:
At Amazon.com

The pages 490-492 which speaks about convergence criteria for iterated exponentials can be read there by LookInside, but do not contain the formulas mentioned by Gottfried.

Ivars
05/27/2008, 09:35 PM
Post: #7
 Gottfried Long Time Fellow Posts: 731 Joined: Aug 2007
RE: Iteration by Ramanujan
Ivars Wrote:mentioned by Gottfried.

... just cited. I don't know anything about them (don't have JSTOR-access either). Source is the poster galaethea...

Gottfried Helms, Kassel
05/28/2008, 03:47 AM
Post: #8
 andydude Long Time Fellow Posts: 505 Joined: Aug 2007
RE: Iteration by Ramanujan
Ivars Wrote:One place Ramanujan considers infinite exponentials is Notebook 5:
At Amazon.com
Yes indeed, this book can also be found on http://scholar.google.com/ but sadly they require money or something for pages 410 and 490 which are where Ramanujan's iterated exponential formulas are... but luckily the surrounding pages talk about how Bachman recently proved this formula to be true. Bachman's article can be found here.

Andrew Robbins
05/28/2008, 06:52 AM
Post: #9
 bo198214 Administrator Posts: 1,366 Joined: Aug 2007
RE: Iteration by Ramanujan
andydude Wrote:...but luckily the surrounding pages talk about how Bachman recently proved this formula to be true.

Haha, ya Ramanujan was the guy who wildly wrote down a lot a formulas, which mostly could be shown to be true.
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