05/28/2008, 08:10 AM
Code:
bruce berndt's release of ramanujan's notebooks
is one of the best resources for aspiring combinalgebraicists
ramanujan really was one of the greats
and it is a shame that books like "the man who knew infinity"
really do not shine light on ramanujan's methods
if you really want to know what fascinated ramanujan
bruce berndt does a much better job of describing it
^..^
i was recently asked to join a tetration discussion group at http://math.eretrandre.org
and it was mentioned that some of the ramanujan references i have posted may be new to the group
i just wanted to describe
to the best of my knowledge
after a long love affair with ramanujan's notebooks
ramanujan's interest and approach to what is now called tetration
ramanujan has a trait that i strongly admire
he likes to break things
by looking at them bigger than they are
if it applies in one place
and he doesn't immediately see why it can't apply everywhere
then he assumes it applies everywhere
and sees where it takes him
he gets a new tool and starts using it everywhere
ramanujan regularly took discrete relations and looked for continuous generalisations
that was one of his big contributions to number theory
in his quarterly notes
he explores the whole nature of continuous iteration
but elsewhere he shows fascination with the exponential in particular
this is natural since the exponential is fundamental here
in chapter 3 of his notebooks
just after having found out about lagrange inversion
(you know
that point in the budding combinalgebraicists education
where they learn how to invert x = y e^y
just as lambert did so many years ago)
ramanujan suddenly focuses in on
x
x = a e (|a| <= e)
and shows
oo
--- j-1
x \ (j+1) j
e = / -------- a
--- (1)
j=0 j
not much
kinda one of those exercises after learning lagrange inversion
where you substitute ln y
but then he sees the secret to taking another iterate
looks in on
x
e = a + x (a >= 1)
and shows
oo
x --- j-1
e \ (j+1) -aj
e = / -------- e
--- (1)
j=0 j
that type simple and beautiful creation is always very signature of ramanujan
and of course
the next step is the easy for him
.
.
x oo
x --- j-1
x \ (j+1) j
x = / -------- (ln(x)) |ln(x)| < 1/e
--- (1)
j=0 j
at this point in the notebook
berndt lists a huge list of resources on the history of these types of forms
including papers on the convergence of
.
.
.
x3
x2
x1
ramanujan goes on to generalise the whole apparatus in the rest of chapter 3
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
notice that the second sum is a series in terms of r
which can be taken as a real number
he proves some properties about the f_j like
n f (x) = f (x) f' (x)
n 1 n-1
and
oo
---
\
f'(x) = x + / B f (x)
1 --- n n
j=0
(where B_n is the nth bernoulli)
among all sorts of other beautiful theorems and evaluations of f
this expansion of iterated exponentials in continuous r
and the subsequent discovery of many properties of the coefficient functions
is in my opinion ramanujan's largest contribution to tetration
or iterated exponentiation
or whatever you want to call it
although
http://en.wikipedia.org/wiki/Tetration
says that a complex extension of tetration has not been shown to exist
it seems straightforward
(using the expansion of f as
oo
/ x \n ---
f (x) = | - | \ j-1 j
n \ 2 / / (-1) psi (n) x
--- j
j=1
by ramanujan)
that the expansion in f has a positive radius of convergence in r
for a given x < 1
berndt concludes the section with this little bibliographical note
that i have not seen on pages from the tetration community
"I. N. Baker [1][2] has made a thorough study of iterates of entire functions
with particular attention paid to the exponential function in his second
paper. These papers also contain references to work on iterates of
_arbitrary_complex_order_. But we emphasise that no one but Ramanujan
seems to have made study of the coefficients phi_j(r) and f_j(x). A
continued development of this theory appears desirable."
how is that for motivating a student?
now
i personally don't think there is much of a mystery to tetration
i think it's actually a field with quite a history and literature
and i've followed up on many of the leads from berndt
(bell, carlitz, becker, riordan, ginsburg, stanley, ...)
so i've seen some great work in the area
but i suspect there may be a community with interest in these things
who may not be aware of some of this other work
i've tried to point this out on several occasions in the past
but the recent questions i received
from several sources
and interest by the community mentioned above
shows i may need to be more explicit
i will post this to the community forums
and can try to answer questions if any additional references are needed
but i have not pursued tetration in my own studies
instead
except for an early fascination with x^x, x^(-x), and inversion of y^y=x
when i've looked at iterates
i've mostly looked at iterates of other entire functions
/ |0 x \
like G (x) = | | e | - 1
n r+1 \ |n /
and the related lagrange inversion problems on the generalised coshinusi
where you can take ramanujan's work and immediately reapply it...
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galathaea: prankster, fablist, magician, liarsorry for formatting and spaces eaten by forum
).