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
\( \operatorname{sexp}(z)=\operatorname{superf}(z+\theta(z)) \)
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
