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 Equations for Kneser sexp algorithm sheldonison Long Time Fellow Posts: 684 Threads: 24 Joined: Oct 2008 08/08/2010, 08:29 PM (This post was last modified: 08/09/2010, 12:41 AM by sheldonison.) Continuing on, one more quick post today. We have the following equation for sexp(z). $\operatorname{sexp}(z)=\operatorname{superf}(z+\theta(z))$ If we substitude in the equation for $\theta(z) = \operatorname{RiemannCircle}(e^{2\pi i z})$, then we get the following equation for sexp(z), which is only defined for imag(z)>=0. $\operatorname{sexp}(z)=\operatorname{superf}(z+ \operatorname{RiemannCircle}(e^{2\pi i z}))$ But supposed we want the Taylor series for sexp(z), centered at z=0? This particular equation for sexp(z) is only valid for imag(z)>=0. By the Schwarz reflection theorem, for imag(z)<0 sexp(conj(z)) = conj(sexp(z)) Now, since we have defined sexp(z) on for the entire complex plane. We can generate sexp(z) for a unit circle, centered around z=0. Then generate the Taylor series for sexp(z) using the Cauchy Integral theorem. So, in summary, earlier I gave the equation for how to generate the Taylor series for the RiemannCircle(z) from the sexp(z) function. And now I have given a different reverse algorithm, whicch uses the Schwarz reflection property to generate a Taylor series for the sexp(z) function from the RiemannCircle(z) function. In the fast Kneser sexp code post, the pari-GP subroutine for riemaprx(z) is very close to the algorithm I just gave for generating the sexp(z) for imag(z)>0. The loop(n) routine takes those values from the riemaprx, and uses them to generate the updated Taylor series for the sexp(z) function. - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread Equations for Kneser sexp algorithm - by sheldonison - 08/08/2010, 07:14 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 08/08/2010, 08:29 PM RE: Equations for Kneser sexp algorithm - by Gottfried - 08/08/2010, 10:01 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 08/09/2010, 06:27 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 08/11/2010, 06:11 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 08/12/2010, 04:43 AM Riemann mapping, for some scenarios - by sheldonison - 08/12/2010, 03:35 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 08/13/2010, 07:39 PM RE: Equations for Kneser sexp algorithm - by tommy1729 - 08/12/2010, 08:37 PM RE: Equations for Kneser sexp algorithm - by bo198214 - 08/15/2010, 06:34 AM RE: Equations for Kneser sexp algorithm - by sheldonison - 08/15/2010, 10:05 AM RE: Equations for Kneser sexp algorithm - by bo198214 - 06/10/2011, 08:48 AM RE: Equations for Kneser sexp algorithm - by sheldonison - 06/10/2011, 01:43 PM RE: Equations for Kneser sexp algorithm - by bo198214 - 06/13/2011, 01:12 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 06/14/2011, 03:00 PM RE: Equations for Kneser sexp algorithm - by bo198214 - 06/14/2011, 05:07 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 06/19/2011, 03:14 AM RE: Equations for Kneser sexp algorithm - by bo198214 - 06/20/2011, 09:22 PM RE: Equations for Kneser sexp algorithm - by sheldonison - 06/21/2011, 01:48 AM

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