Quote:Your question concerning the selfroot, if intended as the b-solution of y = b^y, is crucial. Either this solution is exclusively given by b = selfrt(y) = y^(1/y), and in this case we have to explain why the "yellow zone" appears, or the b-solution of y = b^y is not the selfroot alone, but it is accompanied by other "functions" or branches. In the second case the explicitation (extraction of b) in y = b^y would have (at least) two branches, the selfroot and the perimeter of the yellow zone. Its inverse, as I see it, must have (... at least) four branches, as it is shown by a "wild" graphical inversion.
GFR
I just thought that given the fact that each nth odd root of a real y has 1 real and n-1 conjugate complex values, we can also study substitution:
y= nth odd root of X; Then selfroot becomes
b= (nth odd root of X ) of (nth odd root of X).
If n =3 we automatically get at least complex 4 values of b in addition to 1 real:
1) b = (a+id ) 3rdroot X) of (( a+id) 3rdroot X)
2)b = (a-id ) 3rdroot X) of (( a+id) 3rdroot X)
3) b = (a+id ) 3rdroot X) of (( a-id) 3rdroot X)
4) b = (a-id ) 3rdroot X) of (( a-id) 3rdroot X)
5) b= real 3rd root(X) of real 3rd root of X.
for n=5, we will have more additonal complex selfroots for b if n=7, etc.
Since any real y is nth odd root of something, those n-1 complex values of y for EACH n are always there, ready for usage, if needed.
Odd numbers are good as these roots does not involve negative numbers, but even in case of a negative root, it can always be expressed as multiplication of 2 imaginary subroots, so in the end there is also 1 real root and n imaginary roots = odd number totally:
e.g square root of X is:
+ sgrt x
- sgrt x = -i*4th real root(X)* -i* 4th real root(X) = +i*4th real root(X)*i* 4th real root(X)
so again, we have 3 roots, whose combinations create different selfroots of Y=sqrt(X) (somehow).
sgrtx
-I*4th real root(X)
+i* 4th real root(X)
Of course, this can be extended infinitely, but may that is the whole point? That every finite real number is a finite root of some other number, meaning it has infinite number of invisible complex "companions".
Ivars
