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