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Infinite tetration of the imaginary unit
#1
We know that the infinite tetration of (applied to) a base b can be found by the pricipal branch of the Lambert Function, by:

y = W(-ln(b))/(-ln b) = h(b).

If we "force" this formula, applying it to base i (the imaginary unit), we get:

h(i) = W(-ln(i))/(-ln(i)) = W(-i * Pi/2) / (-i * Pi/2) , i. e. :

h(i) = i[4]oo = 0.438283.. + i * 0.360592..
See also: http://mathworld.wolfram.com/PowerTower.html formula 18

It is interesting to remember that:
h(e^(Pi/2)) = (e^(Pi/2))[4]oo = {-i,+i}

and also that:
i[4]2 = e^(-Pi/2)

GFR

Corrected on 19th-02-08: (e^(Pi/2))[4]oo = -i (Thanks, Ivars !)
We get +i with another branch of the W formula (W(-1), instead of W(0)). I check again.
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Messages In This Thread
Infinite tetration of the imaginary unit - by GFR - 02/10/2008, 12:09 AM

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