Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
" tommy integreable " (?)
#1
Consider a C^oo function that is nowhere analytic.

Now any Taylor series has radius 0.

And a Taylor series expanded at (any) point A is unrelated to a Taylor series expanded at a(any) different point B.

In other words a (divergent) Taylor series for a C^oo function does NOT uniquely define that C^oo function.

( If this confused you remember that you could add a (C^oo) function that is flat for x<0 and increasing for x>0. If you expand your taylors at -1 and +1 ... Also f(x+v) does not work when expaned at x because the radius = 0 )

Thats intresting.

But are there properties between analytic and C^oo ?

If for all x,y in the domain where f is C^oo , and x,y are connected by a C^oo path : the taylor for f(x) corresponds to the Taylor for f(y) in the sense that :

integral from 0 to x of f(z) dz = F(x).

the Taylor of F ' (x) = the Taylor of f(x).

And the same is true for any y ( x replaced by y ).

Maybe this is Always the case ??

If it is not Always the case , I call this " tommy integreable ".

Why, you might wonder.

Well, I wanted to integrate sexp^[1/2](x) based on my 2sinh method.

So that is where this is all coming from.

Basicly trying to integrate C^oo functions.

Any other ideas are welcome.

regards

tommy1729


Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  " tommy quaternion " tommy1729 12 961 02/14/2021, 02:24 AM
Last Post: JmsNxn
  tommy's simple solution ln^[n](2sinh^[n+x](z)) tommy1729 1 4,057 01/17/2017, 07:21 AM
Last Post: sheldonison
  Tommy's matrix method for superlogarithm. tommy1729 0 2,640 05/07/2016, 12:28 PM
Last Post: tommy1729
  Dangerous limits ... Tommy's limit paradox tommy1729 0 2,839 11/27/2015, 12:36 AM
Last Post: tommy1729
  Tommy's Gamma trick ? tommy1729 7 10,045 11/07/2015, 01:02 PM
Last Post: tommy1729
  Tommy triangles tommy1729 1 3,237 11/04/2015, 01:17 PM
Last Post: tommy1729
  Tommy-Gottfried divisions. tommy1729 0 2,487 10/09/2015, 07:39 AM
Last Post: tommy1729
  Tommy's hyperlog tommy1729 0 2,611 06/11/2015, 08:23 AM
Last Post: tommy1729
Sad Tommy-Mandelbrot function tommy1729 0 2,951 04/21/2015, 01:02 PM
Last Post: tommy1729
  tommy equation tommy1729 3 6,033 03/18/2015, 08:52 AM
Last Post: sheldonison



Users browsing this thread: 1 Guest(s)