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 Taylor polynomial. System of equations for the coefficients. marraco Fellow Posts: 100 Threads: 12 Joined: Apr 2011 05/07/2015, 09:45 AM (This post was last modified: 05/07/2015, 10:29 AM by marraco.) 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 b.(x+1)=b.Σbᵢ.xⁿ which translates into P.[bᵢ]=b.[bᵢ] or [P-b.I].[bᵢ]=0 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. « Next Oldest | Next Newest »

 Messages In This Thread Taylor polynomial. System of equations for the coefficients. - by marraco - 04/30/2015, 03:24 AM RE: Taylor polinomial. System of equations for the coefficients. - by tommy1729 - 05/01/2015, 08:37 AM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/01/2015, 09:42 AM RE: Taylor polinomial. System of equations for the coefficients. - by tommy1729 - 05/01/2015, 09:43 PM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/03/2015, 04:46 AM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/03/2015, 12:07 PM RE: Taylor polinomial. System of equations for the coefficients. - by Gottfried - 05/05/2015, 07:40 AM RE: Taylor polinomial. System of equations for the coefficients. - by marraco - 05/06/2015, 02:42 PM RE: Taylor polinomial. System of equations for the coefficients. - by Gottfried - 05/06/2015, 04:17 PM RE: Taylor polynomial. System of equations for the coefficients. - by marraco - 05/07/2015, 09:45 AM RE: Taylor polynomial. System of equations for the coefficients. - by marraco - 01/14/2016, 12:47 AM

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