Let f(x) be a real-entire function with all derivatives > 0 and f(0) >= 0.

Let C be the Cauchy constant ( 1/ 2 pi i ).

The taylor coëfficiënts are given by the contour integral

[1]

C *

The estimate from fake function theory,

Min (f(x) x^{-n}) can also be given by a contour integral

Let g(x,n) = f(x) / x^n.

then

[2]

Min (f(x) x^{-n}) = C *

So the correcting factors are given by

Cor(n) = [1]\[2] =

So the question becomes to estimate , bound or simplify [1]\[2].

Not sure how to proceed here.

But now we have a reformulation in terms of more standard calculus ; in terms of (contour) integration.

I call this the " ratio formulation " and TPID 17 can be expressed in it.

Im aware I did not mention alot of related things such as the specification of the contours , numerical methods , Laplace etc etc.

Certainly special cases can be solved but a general idea is missing.

I was able to prove / disprove the expressibility in similar cases , but contour integration is a bit trickier then " ordinary " integrals.

Ideal would be if we could express this ratio as a single contour.

But im not sure if that is possible.

While considering that, the idea of

" contour derivative " [1]\[2]

Comes to mind.

For Some of you - or most - this was already clear I assume.

But for completeness I make this post.

Also Sheldon has similar ideas and I am not sure how exactly they relate ...

Regards

Tommy1729

Let C be the Cauchy constant ( 1/ 2 pi i ).

The taylor coëfficiënts are given by the contour integral

[1]

C *

The estimate from fake function theory,

Min (f(x) x^{-n}) can also be given by a contour integral

Let g(x,n) = f(x) / x^n.

then

[2]

Min (f(x) x^{-n}) = C *

So the correcting factors are given by

Cor(n) = [1]\[2] =

So the question becomes to estimate , bound or simplify [1]\[2].

Not sure how to proceed here.

But now we have a reformulation in terms of more standard calculus ; in terms of (contour) integration.

I call this the " ratio formulation " and TPID 17 can be expressed in it.

Im aware I did not mention alot of related things such as the specification of the contours , numerical methods , Laplace etc etc.

Certainly special cases can be solved but a general idea is missing.

I was able to prove / disprove the expressibility in similar cases , but contour integration is a bit trickier then " ordinary " integrals.

Ideal would be if we could express this ratio as a single contour.

But im not sure if that is possible.

While considering that, the idea of

" contour derivative " [1]\[2]

Comes to mind.

For Some of you - or most - this was already clear I assume.

But for completeness I make this post.

Also Sheldon has similar ideas and I am not sure how exactly they relate ...

Regards

Tommy1729