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 Observations on power series involving logarithmic singularities bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 11/06/2007, 12:46 PM (This post was last modified: 11/07/2007, 11:05 AM by bo198214.) Though the question was not to me, the matrix form of $f(e^x)=f(x)+1$ or by composition $f\circ\exp = \tau_1\circ f$ is $E\cdot F = F\cdot T_1$ where E is the Carleman matrix of exp and so on. $T_1$ is the upper triangular pascal matrix. But this does not help here/gives no way to solve the equation. If we however reduce F to its first column $(f_0,f_1,\dots)^T$ on the left side, we can reduce $T_1$ to the first column on the right side $t_1=(1,1,0,\dots)^T$. Then $F\cdot t_1=(1+f_0,f_1,f_2,\dots)^T=\vec{f}+(1,0,\dots)^T$ and the fate is on its way $E\cdot\vec{f}=\vec{f}+(1,0,\dots)^T$ and $(E-I)\vec{f}=(1,0,\dots)^T$. For the truncated matrices I think for no $F$ is $E=F\cdot T_1 \cdot F^{-1}$ because the Jordan normal form of $E$ is a Diagonal matrix with pairwise different eigenvalues while the Jordan normal form of $T_1$ has only the eigenvalue 1. « Next Oldest | Next Newest »

 Messages In This Thread Observations on power series involving logarithmic singularities - by jaydfox - 10/26/2007, 11:09 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/26/2007, 11:32 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/26/2007, 11:41 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/27/2007, 06:32 AM RE: Observations on power series involving logarithmic singularities - by Gottfried - 10/29/2007, 11:30 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/29/2007, 05:37 PM RE: Observations on power series involving logarithmic singularities - by Gottfried - 10/30/2007, 06:29 AM RE: Observations on power series involving logarithmic singularities - by andydude - 10/29/2007, 11:50 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/30/2007, 02:25 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/30/2007, 03:40 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/30/2007, 05:33 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 10/31/2007, 08:55 PM RE: Observations on power series involving logarithmic singularities - by Gottfried - 11/03/2007, 06:02 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 07:28 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/05/2007, 11:08 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 02:28 PM RE: Observations on power series involving logarithmic singularities - by Gottfried - 11/06/2007, 07:29 AM RE: Observations on power series involving logarithmic singularities - by Gottfried - 11/06/2007, 11:51 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/06/2007, 12:46 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 07:42 AM RE: Observations on power series involving logarithmic singularities - by andydude - 11/05/2007, 08:20 AM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 02:19 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/05/2007, 02:33 PM RE: Observations on power series involving logarithmic singularities - by andydude - 11/10/2007, 06:19 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/10/2007, 12:46 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/10/2007, 06:02 PM RE: Observations on power series involving logarithmic singularities - by jaydfox - 11/11/2007, 01:01 AM RE: Observations on power series involving logarithmic singularities - by andydude - 11/12/2007, 08:26 AM RE: Observations on power series involving logarithmic singularities - by andydude - 11/12/2007, 08:34 AM RE: Observations on power series involving logarithmic singularities - by bo198214 - 11/12/2007, 10:59 AM

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