If we work with this:
W(-x*(pi/2) - I*x*ln(x) ) = ln(x) - I/pi/2) for x>1
by substituting y=(pi/2) -I*ln(x)), y> pi/2 we get lnx = -I*y+I*pi/2 and x= e^ln(x) = I*e^(-I*y)
W(-(I*y)*e^(-I*y)) = -Iy ; y>pi/2
then infinite tetration of any complex number that is a self root of e^(I*y) is :
h(e^(I*y)^e(-I*y)) = (e^(I*y)) this works for y>=pi/2, at least.
Its no big help, since to find y anyway Lambert function is needed, but if we just vary y, we will get argument and h for each y.
since e^I= Om^(-I/Om), where Om=Omega constant, 0,567143... it is also true that:
h((Om^((-I*y/Om))^((Om)^(I*y/Om)))) =h((Om^(-I*y*Om))^(I*y/Om-1))= Om^((-I*y)/Om))
We can see that result is purely imaginary with values +-i only if y= pi/2, 3pi/2, 5pi/2 etc. Than means that Gottfrieds spider graphs values +- i oscilate at:
e^I*n*pi/2^e^(-I*n*pi/2) for n>=1 and odd or:
h(i^(1/i)) = h(4,81407738096535) = e^(I*pi/2)= I
h(i^3^(1/i^3)) =h(4,81407738096535) = e^(3pi/2) = -I
h(i^5^(1/i^5) =h(4,81407738096535) = e^(5pi/2) = I
h((i^7)^(1/i^7) = h(4,81407738096535)= e^(7pi/2) = -I etc.
Purely Real values of h happen at y= pi/2*k, where k = 2,4,6....
h(e^(Ipi)^(e^(-Ipi))) = h( -1) = e^Ipi= -1
h(e^(2Ipi)^(e^(-2I*pi))=h(1) = e ^2pi= 1
h(e^(3Ipi)^(e^(-3I*pi))=h(-1) = e^3pi = -1
h(e^(4Ipi)^(e^(-4*pi))= h(1) = e^4pi= 1 etc.
It is interesting is there a formula for y<pi/2 which means argument for h with fractions of I.
I hope there were not too many mistakes, but if there were, please help me to find them.
Ivars
W(-x*(pi/2) - I*x*ln(x) ) = ln(x) - I/pi/2) for x>1
by substituting y=(pi/2) -I*ln(x)), y> pi/2 we get lnx = -I*y+I*pi/2 and x= e^ln(x) = I*e^(-I*y)
W(-(I*y)*e^(-I*y)) = -Iy ; y>pi/2
then infinite tetration of any complex number that is a self root of e^(I*y) is :
h(e^(I*y)^e(-I*y)) = (e^(I*y)) this works for y>=pi/2, at least.
Its no big help, since to find y anyway Lambert function is needed, but if we just vary y, we will get argument and h for each y.
since e^I= Om^(-I/Om), where Om=Omega constant, 0,567143... it is also true that:
h((Om^((-I*y/Om))^((Om)^(I*y/Om)))) =h((Om^(-I*y*Om))^(I*y/Om-1))= Om^((-I*y)/Om))
We can see that result is purely imaginary with values +-i only if y= pi/2, 3pi/2, 5pi/2 etc. Than means that Gottfrieds spider graphs values +- i oscilate at:
e^I*n*pi/2^e^(-I*n*pi/2) for n>=1 and odd or:
h(i^(1/i)) = h(4,81407738096535) = e^(I*pi/2)= I
h(i^3^(1/i^3)) =h(4,81407738096535) = e^(3pi/2) = -I
h(i^5^(1/i^5) =h(4,81407738096535) = e^(5pi/2) = I
h((i^7)^(1/i^7) = h(4,81407738096535)= e^(7pi/2) = -I etc.
Purely Real values of h happen at y= pi/2*k, where k = 2,4,6....
h(e^(Ipi)^(e^(-Ipi))) = h( -1) = e^Ipi= -1
h(e^(2Ipi)^(e^(-2I*pi))=h(1) = e ^2pi= 1
h(e^(3Ipi)^(e^(-3I*pi))=h(-1) = e^3pi = -1
h(e^(4Ipi)^(e^(-4*pi))= h(1) = e^4pi= 1 etc.
It is interesting is there a formula for y<pi/2 which means argument for h with fractions of I.
I hope there were not too many mistakes, but if there were, please help me to find them.
Ivars
