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 Observations on power series involving logarithmic singularities jaydfox Long Time Fellow Posts: 440 Threads: 31 Joined: Aug 2007 10/27/2007, 06:32 AM jaydfox Wrote:Then just today, I was thinking about the sexp function, having finally decided to turn my attention back to it a couple days ago. Anyway, I was thinking about the singularity at sexp(-2). It occurred to me that in the immediate vicinity of -2, it would look pretty much like $\ln(a_1 z)$, where $a_1$ is the coefficient for the first degree term in the power series for sexp. In fact, this would just reduce to $\ln(z)+\ln(a_1)$. As such, near the singularity, and assuming no other singularities in the immediate vicinity, the power series for sexp should approximately equal the power series for the natural logarithm. Sure enough, I took my terms for the power series of the slog at 0, and calculated the reversion of the series to get the power series for the sexp (at -1). When I calculated the power series for the first derivative (a trivial calculation), I found that after the first half dozen terms, the terms of the first derivative of the sexp are alternating plus or minus 1 to within 1% or less, and by the 18th term, they're equal to +/- 1 to within 1 part in a million. In other words, the terms of the power series of sexp(z-1) converge on the terms for the power series of ln(z+1).I suppose this isn't so interesting if you take the power series for sexp(z-1). After all, if you assumed a generic linear critical interval (-1, 0), or even a simple third order approximation, then the interval (-2, -1) would pretty much look like a logarithm, and as such, the power series at -1 (from the left) would pretty much be that of a logarithm. But where it would get interesting is when you take power series further and further to the right. The power series at z=0, z=1, z=2, etc., would start out looking more and more like iterated exponentials, yet they would still converge on the power series of a logarithm with its singularity at -2. ~ Jay Daniel Fox « 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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