06/07/2011, 05:54 PM
(This post was last modified: 06/07/2011, 08:59 PM by sheldonison.)
(06/07/2011, 05:47 PM)bo198214 Wrote: Wow, Sheldon, thats such a great overview!I have to update that typo in my picture. Im(f(z))~=0 except at the real axis. For the NewFunc (Kneser mapping), Im(f(z))~=0 means that Im(f(z)) is wobbling very close to zero, but not equal to zero.
Only one clarification: What means ~=? I guess it means not equal.
but then your "new function" Im(z)=8.6 must read "Im(f(z))=0" instead of ~=0?
It must be symmetric about the real axis.
Quote:But I am really amazed how quickly you develop your code for new functions!I haven't calculated the Fatou function yet, but now that I have drawn the pictures, it should be easy to calculate (easy compared to a Kneser Riemann mapping), and convergence precision should improve much much quicker than the Kneser Riemann mapping. It will still be an iterative mapping, iteratively generating each of the two theta(z) functions from the other superfunction, and the other theta(z) function. Because there is so much overlap where both theta(z) functions are analytic, no intermediate "sexp" function will be required to generate the mapping.
Say, is your "new function" analytic continuable through base eta?
Yes, the Kneser mapping "newfunc", for B<eta is continuable through to eta, if you provide a corresponding continuous sequence of upper superfunctions, from B=sqrt(2) to B=eta. Of course, the imag period goes to infinity at B=eta. In each of these upper superfunctions, there will be suitable contour which can be Kneser mapped, to generate a real valued function for z>-2, which will have singularities at z=-2,-3,-4 .....
Thanks for the compliments.
- Shel