03/24/2023, 12:49 AM
A variant to the 2sinh method.
Instead of using 2sinh(x) we use
f(x) = exp(x) - exp(-x) + exp(-x/2) - exp(-x/5) + exp(-x/6) - exp(-x/9) + exp(-x/10) - ...
The pattern is clear ;
f(x) = exp(x) - exp(-x) + [exp(-x/2) - exp(-x/5)] + [exp(-x/6) - exp(-x/9)] + [exp(-x/10) - ...
Notice that this also has a fixpoint at 0.
exp(x) - exp(-x) + exp(-x/2) - exp(-x/5) + exp(-x/6) - exp(-x/9) + exp(-x/10) - ...
is equal to
2sinh(x) + sum [ exp( -x/(4n + 2) ) - exp( -x/(4n + 5) ) ]
where the sum over n goes from n=0 to n = oo.
This is an infinite correction from 2sinh(x) towards exp(x).
Notice that every R ' th derivative of f(x) for x>0 is larger than 0.
Where R is a positive real not just a positive integer.
So the approximation is very smooth.
The first derivative at x = 0 is
2 + sum [ 1/(4n + 2) - 1/(4n + 5) ] = 3 - pi/8 - ln(2)/4.
or approximately 2.43401...
Since 2 < 2.43 < e
We have the fact that this methods works better for smaller bases then the 2sinh method.
Indeed while the 2sinh method works for bases up to exp(1/2) [=1.648...] , this method works for bases up to exp(1/(2.43..)) [= 1.508...]
which is only slightly above 3/2.
As you probably noticed the method is designed with a few ideas in mind.
1)
It is a better asymptotic for Re(z) > 1 , again on a half-plane.
Notice erf type methods( approximations ) would converge faster but not be asymptotic on a half-plane.
2)
The taylor series at 0 has rational coefficients for every truncated sum for f(x).
3)
The taylor series at 0 has coefficients that can be expressed by factorials and zeta functions.
4)
every R ' th derivative of f(x) for x>0 is larger than 0.
5)
The approximation is smooth and the function f(x) itself is entire.
6)
The idea is not complicated.
7)
Carleman matrices can be used.
Now this might just be a speed up for the 2sinh and it might have ups and downs.
And it might have the semi-group iso.
More research is needed.
fixpoints and zeros etc.
But remember the method is to start with REAL x or carleman matrices around 0.
regards
tommy1729
Truth is what does not go away when you stop believing in it.
Instead of using 2sinh(x) we use
f(x) = exp(x) - exp(-x) + exp(-x/2) - exp(-x/5) + exp(-x/6) - exp(-x/9) + exp(-x/10) - ...
The pattern is clear ;
f(x) = exp(x) - exp(-x) + [exp(-x/2) - exp(-x/5)] + [exp(-x/6) - exp(-x/9)] + [exp(-x/10) - ...
Notice that this also has a fixpoint at 0.
exp(x) - exp(-x) + exp(-x/2) - exp(-x/5) + exp(-x/6) - exp(-x/9) + exp(-x/10) - ...
is equal to
2sinh(x) + sum [ exp( -x/(4n + 2) ) - exp( -x/(4n + 5) ) ]
where the sum over n goes from n=0 to n = oo.
This is an infinite correction from 2sinh(x) towards exp(x).
Notice that every R ' th derivative of f(x) for x>0 is larger than 0.
Where R is a positive real not just a positive integer.
So the approximation is very smooth.
The first derivative at x = 0 is
2 + sum [ 1/(4n + 2) - 1/(4n + 5) ] = 3 - pi/8 - ln(2)/4.
or approximately 2.43401...
Since 2 < 2.43 < e
We have the fact that this methods works better for smaller bases then the 2sinh method.
Indeed while the 2sinh method works for bases up to exp(1/2) [=1.648...] , this method works for bases up to exp(1/(2.43..)) [= 1.508...]
which is only slightly above 3/2.
As you probably noticed the method is designed with a few ideas in mind.
1)
It is a better asymptotic for Re(z) > 1 , again on a half-plane.
Notice erf type methods( approximations ) would converge faster but not be asymptotic on a half-plane.
2)
The taylor series at 0 has rational coefficients for every truncated sum for f(x).
3)
The taylor series at 0 has coefficients that can be expressed by factorials and zeta functions.
4)
every R ' th derivative of f(x) for x>0 is larger than 0.
5)
The approximation is smooth and the function f(x) itself is entire.
6)
The idea is not complicated.
7)
Carleman matrices can be used.
Now this might just be a speed up for the 2sinh and it might have ups and downs.
And it might have the semi-group iso.
More research is needed.
fixpoints and zeros etc.
But remember the method is to start with REAL x or carleman matrices around 0.
regards
tommy1729
Truth is what does not go away when you stop believing in it.