For infinite tetration of z \( z<e^{-e} \):
\( h(z)= 1/x; 1/y \) such that \( z=(1/x)^y = (1/y)^x \)
e.g if x=2, y =4, \( z= (1/2)^4=(1/4)^2=1/16 \) and
\( h(z)=1/2; 1/4 \)
e.g if x=1.78381425177, y =4.89536795553, \( z= (1/1.78381425177)^{4.89536795553}=(1/4.89536795553)^{1.78381425177}=1/17 \) and
\( h(z)=h(1/17)= 1/1.78381425177; 1/4.89536795553 \)
Oops I did not notice Henryk has already written the same in previous post.
I also like the idea that function (filter above) can converge to a multinumber set. This could be applicable also to the strange 3_logarithm I introduced in another thread Generalization of logarithms with property 3_log(z=x^y) = 3_log(x)*3_log(y) , which basically maps ALL values of Real z=x^y to set {0,1}, so for all z it converges to set {0,1}, useful or not.
In case h(b), b< e^-e 2 number set moves with b, according to rule above and more exactly stated by Henryk in previous post. In limit b->0, this set becomes {0,1}. In case of 3-logarithm, it does not move at all at least for real x, and is {0,1}.
so 3_log(z) = lim z->0 h(z) = {0,1} if z is real and h(z) stays purely real.
Ivars
\( h(z)= 1/x; 1/y \) such that \( z=(1/x)^y = (1/y)^x \)
e.g if x=2, y =4, \( z= (1/2)^4=(1/4)^2=1/16 \) and
\( h(z)=1/2; 1/4 \)
e.g if x=1.78381425177, y =4.89536795553, \( z= (1/1.78381425177)^{4.89536795553}=(1/4.89536795553)^{1.78381425177}=1/17 \) and
\( h(z)=h(1/17)= 1/1.78381425177; 1/4.89536795553 \)
Oops I did not notice Henryk has already written the same in previous post.
Quote:One could say that converges to the set . And if you consider convergence of filters in general topology they indeed converge to sets.
I also like the idea that function (filter above) can converge to a multinumber set. This could be applicable also to the strange 3_logarithm I introduced in another thread Generalization of logarithms with property 3_log(z=x^y) = 3_log(x)*3_log(y) , which basically maps ALL values of Real z=x^y to set {0,1}, so for all z it converges to set {0,1}, useful or not.
In case h(b), b< e^-e 2 number set moves with b, according to rule above and more exactly stated by Henryk in previous post. In limit b->0, this set becomes {0,1}. In case of 3-logarithm, it does not move at all at least for real x, and is {0,1}.
so 3_log(z) = lim z->0 h(z) = {0,1} if z is real and h(z) stays purely real.
Ivars