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 * \( \oint x^{-n-1}f(x) \; \; \)
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 * \( \oint g(x,n) g ' ' (x,n)/g ' (x,n) \)
So the correcting factors are given by
Cor(n) = [1]\[2] =
\( \frac{ \oint x^{-n-1}f(x) \; \; }{ \oint g(x,n) g ' ' (x,n)/g ' (x,n)} \)
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 * \( \oint x^{-n-1}f(x) \; \; \)
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 * \( \oint g(x,n) g ' ' (x,n)/g ' (x,n) \)
So the correcting factors are given by
Cor(n) = [1]\[2] =
\( \frac{ \oint x^{-n-1}f(x) \; \; }{ \oint g(x,n) g ' ' (x,n)/g ' (x,n)} \)
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
