07/26/2009, 01:50 AM
(This post was last modified: 07/26/2009, 01:52 AM by Kouznetsov.)
(07/25/2009, 04:38 AM)Tetratophile Wrote: .. find tet z, find inverse, find all the branchpoints...The inverse of tetration is arctetration; some of its Riemann surfaces are plotted at
http://www.ils.uec.ac.jp/~dima/PAPERS/2009fractae.pdf
As for the Riemann surfaces of tetration, they are not so spectacular. I post the complex map of modified tetratin \( f=\mathrm{tet}_{\rm mofidied}(x+iy) \)
levels of constant \( p=\Re(f) \) and those of \( p=\Im(f) \) are shown with thick lines for integer values. In this function, the cut line (dashed) from the point \( -2 \) is directed vertically, to \( -2-\rm i \infty \); the other cut still runs horisontally, left along the real axis.
In the upper halfplane, \( y>0 \), as well as at \( x>0 \), this function coincides with the conventional tetration, plotted previously. Note that the only imaginaty part of \( f \) has jump at the vertical cut. The real part remains continuous at this cut.
Quote:I don't htink the cut at z<-2 of tet z is a branch cut. it is where from all directions the function blows up?Perhaps, you wanted to say "branch point".
The function is not equal to its Taylor series, developed in its branchpoint.
(Even it the series exist.) The function has no need to blow up in vicinity of the branch point. For example, the \( \sqrt{\exp} \) remains bounded in vicinity of its branch points.
Henryk, may I post here the plots of \( \sqrt{\exp} \) and \( \sqrt{!} \) in order to show that a function has no need to blow up in vicinity of its branch point?
