Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Vincent's theorem and sin(sexp) ?
#1
While thinking about the sign changes in the truncated Taylor series for functions such as sin(sinh(x)), sin(e^x) or the more exotic ones such as sin(sexp(x)) , sin(exp^[1/2](x)) i was fascinated by apparant trends.

For instance the signs of the truncated taylors ( ignoring the zero's ) of sin(sinh(x)) and sin(e^x) tend to never have more than 9+'s or 9-'s in a row.
Hence the max amount of sign changes appears to be bounded by n/9 where n is the degree (of the polynomial) of the truncation.

Now I know - as most here - that there are formula's for the Taylor coefficients of f(g(x)) when they are known for f and g.

However that does not make those apparant trends seem trivial.

There is alot written about Taylor series and related series.
There is also written alot about polynomials.
SO I might have forgotten something trivial but I seem to be missing the BIG PICTURE here.

Whenever f(x) grows faster than polynomial , sin(f(x)) is somewhat mysterious to me when it comes to sign changes.

Since sin(exp(x)) or sin(sexp(x)) must have an infinite amount of real zero's it is clear that the zero's of that function relate to the sign changes.

So I was thinking about using the error term theorems about Taylor series together with Vincent's theorem to understand this better.

But Im still a bit confused. And I do not know how to generalize things.

Probably there are some uniqueness conjectures to be made about the sign changes of sin(sexp(x)) too. ( uniqueness for sexp )

Im not sure how " standard " these questions are - like I said I might have forgotten stuff - but if they are classical they seem to be well hidden.

Regards

tommy1729
Reply


Possibly Related Threads...
Thread Author Replies Views Last Post
  Some "Theorem" on the generalized superfunction Leo.W 44 9,179 09/24/2021, 04:37 PM
Last Post: Leo.W
  tommy's singularity theorem and connection to kneser and gaussian method tommy1729 2 449 09/20/2021, 04:29 AM
Last Post: JmsNxn
  Sexp redefined ? Exp^[a]( - 00 ). + question ( TPID 19 ??) tommy1729 0 3,240 09/06/2016, 04:23 PM
Last Post: tommy1729
  Can sexp(z) be periodic ?? tommy1729 2 6,964 01/14/2015, 01:19 PM
Last Post: tommy1729
  Theorem in fractional calculus needed for hyperoperators JmsNxn 5 11,737 07/07/2014, 06:47 PM
Last Post: MphLee
  pseudo2periodic sexp. tommy1729 0 3,124 06/27/2014, 10:45 PM
Last Post: tommy1729
  [2014] tommy's theorem sexp ' (z) =/= 0 ? tommy1729 1 5,065 06/17/2014, 01:25 PM
Last Post: sheldonison
  Theorem on tetration. JmsNxn 0 2,879 06/09/2014, 02:47 PM
Last Post: JmsNxn
  Multiple exp^[1/2](z) by same sexp ? tommy1729 12 24,542 05/06/2014, 10:55 PM
Last Post: tommy1729
  entire function close to sexp ?? tommy1729 8 16,902 04/30/2014, 03:49 PM
Last Post: JmsNxn



Users browsing this thread: 1 Guest(s)