03/27/2015, 11:20 PM
What I meant is that you have tetration !
Not just for the bases between 1 and eta but for all real bases larger than 1.
Not sure if you realise it yet.
Here is a sketchy way to show it :
In short since you can interpolate analyticly x^x^... = m
where ... are integer iterations and m is a real > e ...
You can THUS solve for the x in tet_x(t) = m for a given m > e and t >0.
( x is the base ).
But this also means that you can solve for t since you can set up the equation RAM(m,t) = x for any desired x.
WHen you have this t , you have found tet_base_x(t) = m for a given m.
In other words from the relation tet_x(t) = m you can solve for either x or t.
therefore you can solve sexp_x(t) = m
which is slog_x(m).
Then invert this function and you have sexp for any base > 1.
Since all of this is done analyticly you have found tetration.
And it seems simpler then some other methods , like Kneser or Cauchy.
Hope this is clear enough.
I can explain more if required.
SO JmsNxn finally has his own method , with credit to the brilliant comment of fivexthethird.
( Im thinking of a variant of this method too )
I just wonder what this will be called ...
JMS method ? JN method ? Jms5x3 method ?
Jms5x31729 method
I already started calling it in my head " Ramanujan-Lagrange method ".
The reason seems clear : Ramanujan's master theorem and Lagrange's inversion theorem.
For those unfamiliar :
http://en.wikipedia.org/wiki/Lagrange_inversion_theorem
regards
tommy1729
Not just for the bases between 1 and eta but for all real bases larger than 1.
Not sure if you realise it yet.
Here is a sketchy way to show it :
In short since you can interpolate analyticly x^x^... = m
where ... are integer iterations and m is a real > e ...
You can THUS solve for the x in tet_x(t) = m for a given m > e and t >0.
( x is the base ).
But this also means that you can solve for t since you can set up the equation RAM(m,t) = x for any desired x.
WHen you have this t , you have found tet_base_x(t) = m for a given m.
In other words from the relation tet_x(t) = m you can solve for either x or t.
therefore you can solve sexp_x(t) = m
which is slog_x(m).
Then invert this function and you have sexp for any base > 1.
Since all of this is done analyticly you have found tetration.
And it seems simpler then some other methods , like Kneser or Cauchy.
Hope this is clear enough.
I can explain more if required.
SO JmsNxn finally has his own method , with credit to the brilliant comment of fivexthethird.
( Im thinking of a variant of this method too )
I just wonder what this will be called ...
JMS method ? JN method ? Jms5x3 method ?
Jms5x31729 method
I already started calling it in my head " Ramanujan-Lagrange method ".
The reason seems clear : Ramanujan's master theorem and Lagrange's inversion theorem.
For those unfamiliar :
http://en.wikipedia.org/wiki/Lagrange_inversion_theorem
regards
tommy1729
