Thread Rating:
  • 0 Vote(s) - 0 Average
  • 1
  • 2
  • 3
  • 4
  • 5
Taylor polynomial. System of equations for the coefficients.
I misinterpreted what the Carleman matrix was. I tough that it contained the powers of the derivatives of a function (valued at zero), but it contains the derivatives of the powers of a function, so it actually haves the products of the aᵢ coefficients (of bᵢ in your notation).


I tried to use this method to find the coefficients for exponentiation: bˣ=Σbᵢ.xⁿ

The condition is

which translates into


The solution should be bᵢ=ln(b)ⁱ / i!

I found bᵢ=c. (ln(b)ⁱ/i!), where c is an arbitrary constant, because, obviously c.b⁽ˣ⁺¹⁾=b.(c.bˣ)

I was bugged for the fact that any equation for solving tetration I tried seems to have at least one degree of liberty. I think now that it should be explained by one (at least) arbitrary constant in the solution.

This looks analogous to constants found in the solution of differential equations, so I wonder if the evolvent of the curves generated by the constant is also a solution, and what is his meaning.
I have the result, but I do not yet know how to get it.

Messages In This Thread
RE: Taylor polynomial. System of equations for the coefficients. - by marraco - 05/07/2015, 09:45 AM

Possibly Related Threads...
Thread Author Replies Views Last Post
  Arbitrary Order Transfer Equations JmsNxn 0 321 03/16/2021, 08:45 PM
Last Post: JmsNxn
  New Quantum Algorithms (Carleman linearization) Finally Crack Nonlinear Equations Daniel 2 872 01/10/2021, 12:33 AM
Last Post: marraco
  Moving between Abel's and Schroeder's Functional Equations Daniel 1 2,440 01/16/2020, 10:08 PM
Last Post: sheldonison
Question Taylor series of i[x] Xorter 12 20,796 02/20/2018, 09:55 PM
Last Post: Xorter
  Taylor series of cheta Xorter 13 22,724 08/28/2016, 08:52 PM
Last Post: sheldonison
  Totient equations tommy1729 0 3,023 05/08/2015, 11:20 PM
Last Post: tommy1729
  Bundle equations for bases > 2 tommy1729 0 3,036 04/18/2015, 12:24 PM
Last Post: tommy1729
  Grzegorczyk hierarchy vs Iterated differential equations? MphLee 0 3,207 01/03/2015, 11:02 PM
Last Post: MphLee
  A system of functional equations for slog(x) ? tommy1729 3 7,348 07/28/2014, 09:16 PM
Last Post: tommy1729
  partial invariant equations ? tommy1729 0 2,957 03/16/2013, 12:32 AM
Last Post: tommy1729

Users browsing this thread: 1 Guest(s)