An idea that is very very old.
The connection between series multisection and fake function theory.
An example says more than a 1000 pictures.
Consider f(x) = 1 + x + x^2/2! + x^3/3! - x^4/4! + ...
where the sign pattern continues as +,+,+,- such that every multiple of 4 gives a minus sign.
If you ask someone to estimate f(x) for x > 0 , they will likely say
f(x) ~ 1/4 + 1/2 exp(x) + C for some small real C.
Now the logical questions are
how good is this estimate really ... in other words a deeper study.
Clearly this relates to the mittag leffler function and the classic formula for series multisection that uses roots of unity.
But more relevant here is
fake f(x) ~ 1/4 + 1/2 exp(x) + C ??
How close to the truth is that ?
How good does fake function theory estimate here ?
Is fake function theory the ultimate method for this , or is it weak ?
Also notice the alternative estimates
1 + sinh(x)
or
cosh(x)
Who also have positive derivatives.
---
The differential equations
d^n f / d^n x = f(x)
are also often considered because of the natural connection.
---
These questions seems very reasonable and solvable.
Generalized questions and answers are therefore very likely to exist.
regards
tommy1729
" Together we can do more "
tommy1729
The connection between series multisection and fake function theory.
An example says more than a 1000 pictures.
Consider f(x) = 1 + x + x^2/2! + x^3/3! - x^4/4! + ...
where the sign pattern continues as +,+,+,- such that every multiple of 4 gives a minus sign.
If you ask someone to estimate f(x) for x > 0 , they will likely say
f(x) ~ 1/4 + 1/2 exp(x) + C for some small real C.
Now the logical questions are
how good is this estimate really ... in other words a deeper study.
Clearly this relates to the mittag leffler function and the classic formula for series multisection that uses roots of unity.
But more relevant here is
fake f(x) ~ 1/4 + 1/2 exp(x) + C ??
How close to the truth is that ?
How good does fake function theory estimate here ?
Is fake function theory the ultimate method for this , or is it weak ?
Also notice the alternative estimates
1 + sinh(x)
or
cosh(x)
Who also have positive derivatives.
---
The differential equations
d^n f / d^n x = f(x)
are also often considered because of the natural connection.
---
These questions seems very reasonable and solvable.
Generalized questions and answers are therefore very likely to exist.
regards
tommy1729
" Together we can do more "
tommy1729
