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 ramanujan and tetration bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 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 $e^x$ would look like. But now I am a bit disappointed. What he considers is not iterated $e^x$ but iterated $e^x-1$! 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 $f(0)\neq 0$. Ivars Long Time Fellow Posts: 366 Threads: 26 Joined: Oct 2007 05/30/2008, 12:38 PM (This post was last modified: 05/30/2008, 03:58 PM by Ivars.) galathaea Wrote:in the formal setting jumping to iterating exponentiation misses a whole lot of other growth orders in fact we can start with iterating the original functions the numerator over x and the denominator over factorial and this is key to the generalisation needed because there are many ways to ensure the correct asymptotic order conditions using a variety of iterative techniques to build function orders all of these lie between the realm of the exponential and ramanujan's beast and there is an infinite hierarchy even beyond each waiting for a theory to develop and interesting relations to find Thanks galathaea for answer to my musings and further development. I am interested in tetration ( and further) because it is the next obvisously integer order of infinity. If rules and analogies for discrete enumerable by some integers orders of infinity can be established, than later it should be possible to cover all intermediate ranges by extensions to fractional and rational and real and complex change of orders of infinity as usually. So I am looking to start with integers that enumerate these discrete orders of infinity and then look back into what is between exponentiation and tetration- if needed. I was trying to link it to combinatorics of branching tree chains but can not find any basic text about this subject to even understand the conventions people working in them use. One thought about trees is that divergent series most likely end up in different (almost?) continuous orders of infinity via tree type ( bifurcation, trifurcation, n furcation etc) structure. The correspondance between a type of series and order of infinity ? Ivars « Next Oldest | Next Newest »

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