Thread Rating:
  • 1 Vote(s) - 5 Average
  • 1
  • 2
  • 3
  • 4
  • 5
using sinh(x) ?
#49
(07/18/2010, 10:41 PM)tommy1729 Wrote: prove convergence.

that might sound weird , but my formula uses a limit.

that limit hasnt been proven to converge , it could be a double limit or chaotic or ...

in fact , it is known that exp[n](z) can = oo while exp[n](z+ complex infinitesimal) =/= oo for infinitely many complex z.

secondly

prove that it is continuous.

for similar reasons as above this is not yet proven.

of course on the real line , my formula is both converging and continu but we are considering the complex numbers.

third

of course , prove that it is complex differentiable.

this will probably require the proof of convergence and continuous and wont be provable without them ?

because of the logloglog ... part its taylor series radius must be 0 when expanded at 0.

if you wonder why i believe so , already the first log gives a radius of 0 ?

the idea is this : (truncated) tommysexp(z,x) = ... log log log ( large )

since log has a small radius the large values wont 'fit in' and thus log (large) will have radius 0.

the other logs dont change that : log log ( function with radius 0 ) = function with radius 0.

if im correct ...

im not correct about the last , the log does have a nonzero radius of convergence at least up to the pole of its derivate , hence 1/x = oo -> at x = 0 ; in simpler words : when expanded at point t , the radius is at least abs (t) and thus the taylor series converges in the open set within the circle centered at t with radius abs(t) [ not sure about the convergence on the boundary ]

this generalises to the log iterations in my formula , when considering a finite amount of iterations.

well at least IF Re(x) > 0 and Re(z) > 0.

i think i have a proof that it holds for all the logs ( not just for finite iterations but also for infinite , and -for clarity - not log iterations 'alone' but the WHOLE FORMULA ) and hence my formula is holomorphic , and thus convergeance , continuity and complex differentiability is proven at once.

i will need some time to make the proofs formal or even to post them here , but as you might have already understood , the fast convergence is key.

- assuming the above - what remains to be understood is if my formula satisfies my own condition.

but thats in another thread ... ( called ' tommy's uniqueness conditions ' or similar )

also , numerical problems and acceleration methods need to be considered later on.

contour integration seems usefull and related for proving uniqueness.

i wonder if contour integration is the best way for finding the derivatives numerically.

regards

tommy1729
Reply


Messages In This Thread
using sinh(x) ? - by tommy1729 - 02/28/2010, 12:22 AM
RE: using sinh(x) ? - by tommy1729 - 03/06/2010, 12:35 PM
RE: using sinh(x) ? - by bo198214 - 03/07/2010, 11:19 AM
using 2* sinh(x) ! - by tommy1729 - 03/09/2010, 01:31 PM
RE: using 2* sinh(x) ! - by bo198214 - 03/10/2010, 11:38 AM
RE: using 2* sinh(x) ! - by tommy1729 - 03/11/2010, 12:50 AM
RE: using 2* sinh(x) ! - by tommy1729 - 03/14/2010, 01:07 AM
RE: using 2* sinh(x) ! - by sheldonison - 06/02/2010, 04:47 PM
RE: using 2* sinh(x) ! - by tommy1729 - 06/02/2010, 10:13 PM
RE: using 2* sinh(x) ! - by sheldonison - 06/02/2010, 10:56 PM
RE: using 2* sinh(x) ! - by tommy1729 - 06/07/2010, 12:34 AM
RE: using 2* sinh(x) ! - by sheldonison - 06/08/2010, 01:59 PM
RE: using 2* sinh(x) ! - by tommy1729 - 06/09/2010, 12:18 PM
RE: using 2* sinh(x) ! - by mike3 - 06/11/2010, 02:23 AM
RE: using 2* sinh(x) ! - by bo198214 - 06/12/2010, 04:36 AM
RE: using 2* sinh(x) ! - by mike3 - 06/12/2010, 11:14 AM
RE: using 2* sinh(x) ! - by tommy1729 - 06/21/2010, 09:14 PM
RE: using 2* sinh(x) ! - by sheldonison - 06/23/2010, 06:06 PM
RE: using 2* sinh(x) ! - by tommy1729 - 06/23/2010, 09:11 PM
RE: using 2* sinh(x) ! - by tommy1729 - 06/23/2010, 09:31 PM
RE: using 2* sinh(x) ! - by bo198214 - 06/27/2010, 09:16 AM
RE: using 2* sinh(x) ! - by sheldonison - 06/04/2010, 04:48 AM
RE: using sinh(x) ? - by tommy1729 - 04/24/2010, 11:47 PM
RE: using sinh(x) ? - by bo198214 - 05/31/2010, 05:52 AM
RE: using sinh(x) ? - by tommy1729 - 06/01/2010, 10:58 PM
RE: using sinh(x) ? - by bo198214 - 06/02/2010, 02:58 AM
RE: using sinh(x) ? - by tommy1729 - 06/22/2010, 11:18 PM
RE: using sinh(x) ? - by tommy1729 - 06/23/2010, 10:44 PM
RE: using sinh(x) ? - by tommy1729 - 06/24/2010, 12:29 PM
RE: using sinh(x) ? - by tommy1729 - 06/24/2010, 07:22 PM
RE: using sinh(x) ? - by bo198214 - 06/26/2010, 12:04 PM
RE: using sinh(x) ? - by tommy1729 - 06/26/2010, 09:19 PM
attracting fixed point of 2sinh - by sheldonison - 06/27/2010, 03:31 AM
RE: using sinh(x) ? - by tommy1729 - 07/18/2010, 10:41 PM
RE: using sinh(x) ? - by sheldonison - 07/19/2010, 09:37 PM
RE: using sinh(x) ? - by tommy1729 - 07/20/2010, 08:24 PM
RE: using sinh(x) ? - by tommy1729 - 07/19/2010, 10:20 PM
RE: using sinh(x) ? - by sheldonison - 07/20/2010, 03:13 PM
RE: using sinh(x) ? - by tommy1729 - 07/20/2010, 08:31 PM
RE: using sinh(x) ? - by sheldonison - 07/21/2010, 03:19 AM
RE: using sinh(x) ? - by tommy1729 - 07/21/2010, 10:34 PM
RE: using sinh(x) ? - by sheldonison - 07/22/2010, 12:23 PM
RE: using sinh(x) ? - by tommy1729 - 07/28/2010, 02:24 PM
RE: using sinh(x) ? - by tommy1729 - 07/28/2010, 02:53 PM
RE: using sinh(x) ? - by tommy1729 - 12/04/2010, 11:09 PM
RE: using sinh(x) ? - by tommy1729 - 12/08/2010, 02:31 PM
RE: using sinh(x) ? - by sheldonison - 01/17/2011, 07:43 PM
RE: using sinh(x) ? - by tommy1729 - 01/17/2011, 11:36 PM
RE: using sinh(x) ? - by tommy1729 - 07/02/2011, 10:32 PM
RE: using sinh(x) ? - by tommy1729 - 07/04/2011, 10:56 PM
RE: using sinh(x) ? - by sheldonison - 07/05/2011, 01:09 AM
RE: using sinh(x) ? - by tommy1729 - 07/05/2011, 12:06 PM
RE: using sinh(x) ? - by tommy1729 - 11/18/2012, 11:41 PM
RE: using sinh(x) ? - by sheldonison - 11/20/2012, 12:01 AM
RE: using sinh(x) ? - by tommy1729 - 11/20/2012, 10:53 PM
RE: using sinh(x) ? - by sheldonison - 11/20/2012, 11:01 PM
RE: using sinh(x) ? - by tommy1729 - 11/20/2012, 11:32 PM
RE: using sinh(x) ? - by sheldonison - 11/20/2012, 11:46 PM
RE: using sinh(x) ? - by tommy1729 - 11/28/2012, 06:59 PM
RE: using sinh(x) ? - by tommy1729 - 11/28/2012, 08:07 PM
RE: using sinh(x) ? - by tommy1729 - 12/17/2012, 04:34 PM
RE: using sinh(x) ? - by tommy1729 - 02/10/2015, 12:02 AM
RE: using sinh(x) ? - by tommy1729 - 04/22/2015, 12:27 PM
RE: using sinh(x) ? - by tommy1729 - 04/23/2015, 04:49 PM
RE: using sinh(x) ? - by tommy1729 - 10/26/2015, 12:56 AM
RE: using sinh(x) ? - by tommy1729 - 11/26/2015, 11:56 PM
RE: using sinh(x) ? - by sheldonison - 11/27/2015, 03:41 PM
RE: using sinh(x) ? - by tommy1729 - 11/28/2015, 01:21 PM
RE: using sinh(x) ? - by sheldonison - 11/29/2015, 03:03 PM
RE: using sinh(x) ? - by tommy1729 - 12/01/2015, 01:22 PM
RE: using sinh(x) ? - by sheldonison - 12/01/2015, 02:58 PM
RE: using sinh(x) ? - by tommy1729 - 12/02/2015, 12:04 AM
RE: using sinh(x) ? - by tommy1729 - 12/01/2015, 08:18 PM
RE: using sinh(x) ? - by tommy1729 - 12/02/2015, 12:28 AM
RE: using sinh(x) ? - by tommy1729 - 02/23/2016, 10:36 PM
RE: using sinh(x) ? - by tommy1729 - 02/24/2016, 01:27 PM
RE: using sinh(x) ? - by tommy1729 - 03/01/2016, 10:38 PM
RE: using sinh(x) ? - by tommy1729 - 03/05/2016, 01:27 PM
RE: using sinh(x) ? - by tommy1729 - 03/05/2016, 11:38 PM
RE: using sinh(x) ? - by sheldonison - 03/06/2016, 03:16 PM
RE: using sinh(x) ? - by tommy1729 - 03/06/2016, 10:54 PM
RE: using sinh(x) ? - by tommy1729 - 03/07/2016, 01:48 AM
RE: using sinh(x) ? - by tommy1729 - 03/07/2016, 10:46 AM
RE: using sinh(x) ? - by tommy1729 - 03/07/2016, 09:36 PM
RE: using sinh(x) ? - by tommy1729 - 03/08/2016, 12:26 AM
RE: using sinh(x) ? - by tommy1729 - 03/11/2016, 01:27 PM
RE: using sinh(x) ? - by tommy1729 - 03/12/2016, 01:12 PM
RE: using sinh(x) ? - by tommy1729 - 03/12/2016, 01:20 PM

Possibly Related Threads...
Thread Author Replies Views Last Post
  exp^[3/2](x) > sinh^[1/2](exp(x)) ? tommy1729 7 5,589 10/26/2015, 01:07 AM
Last Post: tommy1729
  2*sinh(3^h*asinh(x/2)) is the superfunction of (...) ? Gottfried 5 5,238 09/11/2013, 08:32 PM
Last Post: Gottfried
  zeta and sinh tommy1729 0 1,912 05/30/2011, 12:07 PM
Last Post: tommy1729



Users browsing this thread: 1 Guest(s)