06/27/2010, 03:31 AM
(This post was last modified: 06/29/2010, 02:02 AM by sheldonison.)
speaking of fixed points, the superfunction of 2sinh has an attracting fixed point of 0 + 1.895494239i, on the imaginary axis. I'm not sure what the exact region of convergence is (probably fractal), but any time real(superfunction(z))=0, then the superfunction(z+1, +2, +3 ... +n) will converge to this attracting fixed point. This is because if real(z)=0, then real(sexp2(z)) will also equal zero, which helps in understand why there is an attracting fixed point.
Regions where real(SuperFunction(z))=0 occur whenever imag(SuperFunction(z))=i*0.5*pi/ln(2), or i*(0.5+n)*pi/ln(2). I started to sketch out where the contours are. The SuperFunction(i*0.5*pi/ln(2)+x) converges to this fixed attracting point as x (real valued) increases. Other imaginary values of z close to 0.5*pi/ln(2) also converge to the attracting fixed point.
This adds complication to the "base change" converting the super function of sexp2 to sexp_e in those regions where real(superfunction(z)) approaches zero, since 2sinh has this attracting fixed point, but exp_e doesn't. Moreover, such regions approach the real axis as z increases.
- Sheldon
Regions where real(SuperFunction(z))=0 occur whenever imag(SuperFunction(z))=i*0.5*pi/ln(2), or i*(0.5+n)*pi/ln(2). I started to sketch out where the contours are. The SuperFunction(i*0.5*pi/ln(2)+x) converges to this fixed attracting point as x (real valued) increases. Other imaginary values of z close to 0.5*pi/ln(2) also converge to the attracting fixed point.
This adds complication to the "base change" converting the super function of sexp2 to sexp_e in those regions where real(superfunction(z)) approaches zero, since 2sinh has this attracting fixed point, but exp_e doesn't. Moreover, such regions approach the real axis as z increases.
- Sheldon