continuous iteration

On May 22, 3:56 pm, Tonico <Tonic...@xxxxxxxxx> wrote:
Also I can't see how that'd help to
solve the question "how can you add 3 to itself 3.54 times".

quite a lot of research has gone into continuous iteration
and it is just as respectable a field as
say
fractional differentiation or fractional dimensions

one common method for sums
(which ramanujan made heavy use of)
is to note that

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

when the right hand sums converge

in the case they converge absolutely
n can be made any value
(i'll call it x to show i'm talking any real here)
and one can "define"

x oo
--- ---
\ \
/ f(j) := / [ f(j) - f(x + j + 1) ]
--- ---
j=0 j=0

there are many similar tricks for other iteration processes

ramanujan has many examples in his quarterly reports
(the essay's he wrote to justify his scholarship)
though even in his much later work
these techniques are often buried in his "proofs"

there is a body of work generally looking at
when extensions to continuous operators is meaningful
when it is unique
when it has multiple solutions or solution spaces
and so on

methods like using matrix representations of the operation
so that exponentiation of the matrix is iteration
are commonly used
and this is explained more deeply in many good combinatorial texts
(like comtet)
as well as in regular posts here by people like gottfried helms

sometimes
as in the case of gamma
you need to add conditions for uniqueness
(like log convexity does with bohr-mollerup)
(and it seems like in your last message

Tetration seems to be a pretty edgy subject,
and since it apparently uses the well known
Gamma function, poor tommy thinks (so to speak)
that if someone doesn't know about tetration
then he also doesn't know about the Gamma
function...:)

you may not be noticing gamma itself is a continuous iterate)

there has never been anything "edgy" about tetration

it has a long history

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galathaea: prankster, fablist, magician, liar
.