05/14/2022, 12:25 PM
about post 26.
considering post 14 and post 18.
First I forgot the constant
sexp(s) = lim ln^[n] f ( s + n + C)
C is neccessary even if f(1) = 1.
So we are not completely rid of constants.
Secondly
I had in mind a function that behaves like t(s) = ( 1 + erf(s))/2 but at integers gives 0 or 1 depending on the sign.
My example gave 0 for 0.
But I wonder about a closed form of an entire function that behaves like t(s) = ( 1 + erf(s))/2 but at integers gives 0 or 1 depending on the sign.
And also that function is desired to behave similar to t(s) on the complex plane in particular when Re(s)^2 > Im(s)^2.
Im not even sure such an entire function exists.
Think of sin(s) growing fast in the imaginary direction.
t(x) sin(v x)^2 + cos(v x)^2 for x > 1 and suitable v does not grow slowly in the complex plane so Im not sure it actually exists with all the desired properties.
Since these properties are lost maybe it does not actually work !?
This requires more study.
However enter post 14 and 18.
We learn from post 14 and 18 the periodic function used for other solutions...
But wait a minute.
The desired function above cannot possible be t(s + p(s)) for a bounded periodic real function p(s).
SO we have a second objection to this modified function.
However clearly the modified function ( or the one where f(1) = 1 from post 26 ) does give a convergent method on the real line.
So here is the main idea :
if t(s) gives an analytic solution , then the modified version does not. Because they cannot both be solutions hence they cannot both be analytic solutions.
A more generalized conjecture is that
F(s) = exp( a(s-1) exp( a(s-2) ... )
can only be analytic if a(s) is
1) going from 0 to 1 on the reals
2) is analytic
AND
3) a(s) is strictly increasing on the reals.
that is mainly post 14 logic talking ...
However the ideas and proofs and conjectures are not formal or decisive yet.
more work is needed.
suppose we take f(s) for our method.
and suppose we take f(2s) for our method.
now we have two methods and two function sexp.
are they equal ?
are they both analytic ? both not ?
by the logic from post 14 f(s) and f(2s) are not related by a periodic function.
but they do both give a solutions on the reals.
and they do satisfy
1) going from 0 to 1 on the reals
2) is analytic
AND
3) a(s) is strictly increasing on the reals.
...
So the issue of acceleration is still a thing ( post 18 ! )
So we find ourselfs again in the realm of weird but interesting conjectures.
regards
tommy1729
considering post 14 and post 18.
First I forgot the constant
sexp(s) = lim ln^[n] f ( s + n + C)
C is neccessary even if f(1) = 1.
So we are not completely rid of constants.
Secondly
I had in mind a function that behaves like t(s) = ( 1 + erf(s))/2 but at integers gives 0 or 1 depending on the sign.
My example gave 0 for 0.
But I wonder about a closed form of an entire function that behaves like t(s) = ( 1 + erf(s))/2 but at integers gives 0 or 1 depending on the sign.
And also that function is desired to behave similar to t(s) on the complex plane in particular when Re(s)^2 > Im(s)^2.
Im not even sure such an entire function exists.
Think of sin(s) growing fast in the imaginary direction.
t(x) sin(v x)^2 + cos(v x)^2 for x > 1 and suitable v does not grow slowly in the complex plane so Im not sure it actually exists with all the desired properties.
Since these properties are lost maybe it does not actually work !?
This requires more study.
However enter post 14 and 18.
We learn from post 14 and 18 the periodic function used for other solutions...
But wait a minute.
The desired function above cannot possible be t(s + p(s)) for a bounded periodic real function p(s).
SO we have a second objection to this modified function.
However clearly the modified function ( or the one where f(1) = 1 from post 26 ) does give a convergent method on the real line.
So here is the main idea :
if t(s) gives an analytic solution , then the modified version does not. Because they cannot both be solutions hence they cannot both be analytic solutions.
A more generalized conjecture is that
F(s) = exp( a(s-1) exp( a(s-2) ... )
can only be analytic if a(s) is
1) going from 0 to 1 on the reals
2) is analytic
AND
3) a(s) is strictly increasing on the reals.
that is mainly post 14 logic talking ...
However the ideas and proofs and conjectures are not formal or decisive yet.
more work is needed.
suppose we take f(s) for our method.
and suppose we take f(2s) for our method.
now we have two methods and two function sexp.
are they equal ?
are they both analytic ? both not ?
by the logic from post 14 f(s) and f(2s) are not related by a periodic function.
but they do both give a solutions on the reals.
and they do satisfy
1) going from 0 to 1 on the reals
2) is analytic
AND
3) a(s) is strictly increasing on the reals.
...
So the issue of acceleration is still a thing ( post 18 ! )
So we find ourselfs again in the realm of weird but interesting conjectures.
regards
tommy1729