10/29/2008, 04:49 AM
(This post was last modified: 11/23/2008, 01:00 AM by Kouznetsov.)

Hello, I did not find figuires of power of exponential. So, I suggest few for .

levels

and

are shown with thick lines in the comples plane

for values (left) and (right).

One of levels coincides with the real axis, but at one of cuts occupies some of the negative part of the real axis. Some intermediate levels are shown with thin lines.

Also, levels and are shown with green curves. These levels cross at the branchpoint

,

which is fixed point of logarithm.

The thick pink lines indicate the cuts of the complex plane.

Graphics are symmetric with respect to the real axis; at real ,

;so, I show only upper half of the complex plane.

Below I incert the graphic for real valuees of , and various values of :

,

,

,

...

-----

After to post the pics, I got the crytics (see the first comment). bo198214 suggests that I describe the method. I expected, that the method is obvious from the name of the forum...

where is holomorpic solution of

The entire soluiton of this equation is described by Kneser [1]; he used it to construct , although it is not the only aplication. Past century, there were no computers, so, Kneser could not plot his solution, and only in this century the plotting becomes possible.

I have implemented tetration "tet", id est, the function , that is holomorphic at least in and bounded at least in

, satisfying conditions

My tetation [2] is holomorphic in the whole complex plane except . I estimate, my algorithm returns at least 14 correct decimal digits; at least, while the argument is of order of unity. Then, I have implemented, with similar precision, its derivative and its interse ; . Such inverse function has two barnchpoints at fixed points and of logarithm, which are solutions of equation . I put the cutlines horizontally, along the halflines , . One of these lines is seen in the figures for at non-ineger .

In order to understand, how does the inverse function work, I plot the image of the upper halfplane with function kslog.

Images of gridlines and are shown. For in the shaded region,

References:

1. H.Kneser. Reelle analytische Losungen der Gleichung und verwandter Funktionalgleichungen''. Journal fur die reine und angewandte Mathematik, v.187 (1950), 56-67.

2. D.Kouznetsov. Solution of in complez z-plane.

Mathematics Of Computations, in press. The up-to-last version is available at my homepage http://www.ils.uec.ac.jp/~dima/PAPERS/analuxp99.pdf

3. http://en.citizendium.org/wiki/Tetration