Improper integrals at MSE

[tommy1729]

Last Friday while playing chess I talked to my MSE friend mick.

You might like this :

http://math.stackexchange.com/questions/...-integrals

regards

tommy1729

[tommy1729]

Sheldon's answer on MSE is nice.

Thank you Sheldon.

I made an intresting observation relating things to complex dynamics.

The main thing is the mysterious looking change ->

My observation can be considered positive or negative , intresting or dissapointing , it depends on taste I guess and the hope for nontrivial analogues.

But the idea of having some function for which every real iterate " works " is found , though it might not be as nontrivial as mick hoped. ( not saying a nontrivial case cannot exist ).

- Maybe variants of this exist in calculus textbooks / papers but its very " dynamical " in nature -

Anyway here it is :





=> with

=>

=>

Solve .. =>

Thus :



Which is trivial.

Reminds me of this quote :

" Young man, in mathematics you don't understand things. You just get used to them. "
John von Neumann.

Btw I considered doing the things (steps above) in reverse : showing -> is valid from the validity of

regards

tommy1729
" the master "

[sheldonison]

....function for which every real iterate " works " is found , though it might not be as nontrivial as mick hoped. ( not saying a nontrivial case cannot exist ).
...




=> with

=>

=>

Solve .. =>

Thus :


....

Hey Tommy,

Not sure I understood all of that ... But it inspired me to consider the following sequence of functions




...


Does g(x) converge, and is it a solution of interest to Mick? If g(x) converges, and it is analytic, then it has a Taylor/Laurent series....

Update:, by brute force, using a lot of computer cycles to estimate the limit, and then turn the coefficents it back into a fraction with power's of 2's... I get the following Laurent series, as the function that Mick might be looking for.


It would probably be normally expressed as


update2:
This would be compactly expressed via the Abel function as:


And then we get:


Finally, Mick's desired function in closed form would be as follows. With a little algebra, we generate all of the fractional iterates of g(z) as well. Then, using Mick's notation we have the desired g(z,t) function, which has all fractional iterates defined as:
for t=1, this is the same as the Laurent series above

[tommy1729]

I once had a few threads here were I discussed the need for limits of the form ( a + f(n)/(bn)) ^[n] = C or similar.

This seems very much like your limit, maybe you got inspired from me too.

Anyways I must say that thread was looking for tetration type functions / limits so in that sense your limit is more " classical ".

I was not able to find the threads again but this one is somewhat similar :

http://math.eretrandre.org/tetrationforu...ight=limit

regards

tommy1729

[tommy1729]

This leads to the question

Is there a solution F(x) = super of ( x + a - 2^t/(x+b) ) for some real a,b (super with respect to t) such that integral f(F(x)) dx = integral f(x) ?

regards

tommy1729

[tommy1729]

I think this is an underrated thread.
If it turns out true , this gives an intresting connection between dynamical systems and calculus !

regards

tommy1729