02/09/2015, 11:59 PM
Very similar to the 2sinh method we can go to bases between eta and exp(1/2).
For instance bases > exp(2/5).
Basicly we just replace 2sinh(x) with
f(x) = 0 + 5/2 x + 5/2 x^5/5! + 5/2 x^6/6! + 5/2 x^10/10!
+ 5/2 x^11/11! + ...
Notice the heart of these ideas ( 2sinh , f(x) above ) is series multisection.
f(A x) is close to exp(A x) , just like sinh(A x).
But f(A x) has derivative at 0 : 5/2 A instead of sinh(A x) which has derivative at 0 : 2 A.
Therefore by using f(A x) we maintain a hyperbolic fixpoint at 0 for all bases > exp(2/5).
Notice f(A x) has all its derivatives >= 0 and 0 =< f(A x) < exp(A x).
The rest is analogue to the 2sinh method.
A little more advanced is the idea that we can VERY LIKELY arrive at a uniqueness criterion regarding these multisection methods.
Although we can express these multisections in terms of standard functions , Im still curious about their " behaviour " such as fixed points , zero's etc.
Some plots would be nice.
( some ideas relating to fake function theory come to mind , but thats way too advanced and speculative for now )
***
The main idea to generalize this towards bases arbitrary close to eta is the simple observation that the multisection is of type "(a,b)" where exp(a/b) is the approximation to eta.
eta ~ exp(a/b)
This naturally makes me wonder about the nicest proofs for the irrationality of eta and the irrationality measure of eta.
It is clear that this method cannot be used for eta itself since
eta =/= exp(a/b)
Unless perhaps some limiting ideas.
Or maybe not since this method is not analytic. Not sure.
***
(musing about the multisection)
HOWEVER
The sky may not be perfectly blue ?
Suppose we work with a multisection type function such as
g(x) = 0 + Q x + Q x^S/S! + ...
Then we need to make sure that
Q x + Q x^S/S! < exp(x)
for all x > 0.
Although Q and S are not indep , the main issue is Q here and a more relaxed equation is
Q x < exp(x)
Therefore we look for the equation
Qx = exp(x)
with a single positive real solution x ;
a tangent line equation.
BUT LUCKILY we find
ex = exp(x) => e = exp(1).
so our Q is exactly bounded by e.
Therefore our base is exactly bounded by eta = exp(1/e).
SOOO , the sky is blue afterall.
THis also proof that the method works for bases larger than eta.
***
Perhaps some intresting links :
http://mathworld.wolfram.com/SeriesMultisection.html
http://mathworld.wolfram.com/IrrationalityMeasure.html
***
I think its true that if
a/b is an approximation of 1/e
then a*/(2b+1) is ALMOST Always a better one.
(where a and a* are chosen optimal ).
this leads to the imho intresting sequence :
( where the denom is iterated under the map 2x+1 )
1/2
2/5
4/11
9/23
17/47
35/95 = 7/19
...
a_n/b_n
and I guess a_(n+1)/b_(n+1) is Always a better approximation then all previous ones.
Its been a while since I did this kind of math so forgive any blunders.
regards
tommy1729
For instance bases > exp(2/5).
Basicly we just replace 2sinh(x) with
f(x) = 0 + 5/2 x + 5/2 x^5/5! + 5/2 x^6/6! + 5/2 x^10/10!
+ 5/2 x^11/11! + ...
Notice the heart of these ideas ( 2sinh , f(x) above ) is series multisection.
f(A x) is close to exp(A x) , just like sinh(A x).
But f(A x) has derivative at 0 : 5/2 A instead of sinh(A x) which has derivative at 0 : 2 A.
Therefore by using f(A x) we maintain a hyperbolic fixpoint at 0 for all bases > exp(2/5).
Notice f(A x) has all its derivatives >= 0 and 0 =< f(A x) < exp(A x).
The rest is analogue to the 2sinh method.
A little more advanced is the idea that we can VERY LIKELY arrive at a uniqueness criterion regarding these multisection methods.
Although we can express these multisections in terms of standard functions , Im still curious about their " behaviour " such as fixed points , zero's etc.
Some plots would be nice.
( some ideas relating to fake function theory come to mind , but thats way too advanced and speculative for now )
***
The main idea to generalize this towards bases arbitrary close to eta is the simple observation that the multisection is of type "(a,b)" where exp(a/b) is the approximation to eta.
eta ~ exp(a/b)
This naturally makes me wonder about the nicest proofs for the irrationality of eta and the irrationality measure of eta.
It is clear that this method cannot be used for eta itself since
eta =/= exp(a/b)
Unless perhaps some limiting ideas.
Or maybe not since this method is not analytic. Not sure.
***
(musing about the multisection)
HOWEVER
The sky may not be perfectly blue ?
Suppose we work with a multisection type function such as
g(x) = 0 + Q x + Q x^S/S! + ...
Then we need to make sure that
Q x + Q x^S/S! < exp(x)
for all x > 0.
Although Q and S are not indep , the main issue is Q here and a more relaxed equation is
Q x < exp(x)
Therefore we look for the equation
Qx = exp(x)
with a single positive real solution x ;
a tangent line equation.
BUT LUCKILY we find
ex = exp(x) => e = exp(1).
so our Q is exactly bounded by e.
Therefore our base is exactly bounded by eta = exp(1/e).
SOOO , the sky is blue afterall.
THis also proof that the method works for bases larger than eta.
***
Perhaps some intresting links :
http://mathworld.wolfram.com/SeriesMultisection.html
http://mathworld.wolfram.com/IrrationalityMeasure.html
***
I think its true that if
a/b is an approximation of 1/e
then a*/(2b+1) is ALMOST Always a better one.
(where a and a* are chosen optimal ).
this leads to the imho intresting sequence :
( where the denom is iterated under the map 2x+1 )
1/2
2/5
4/11
9/23
17/47
35/95 = 7/19
...
a_n/b_n
and I guess a_(n+1)/b_(n+1) is Always a better approximation then all previous ones.
Its been a while since I did this kind of math so forgive any blunders.
regards
tommy1729
