I was playing around with some repeated logarithms for complex values and noticed that it always converged to the same value. A little bit of googling got this result, . I'm a math novice, and I have enjoyed the videos on the Lambert W function. Is there any way to put this explanation into more layman's terms?
Math problem would be: z0 = i
z1 = ln(z0)
.
.
zn = ln(z(n-1))
Per the attached SE site, this converges to .3181315052−1.337235701i . The explanation in the link is a bit over my head, and could use some help getting to a better understanding.
This is because the problem can be formulated as follows:
ln ln ln ln ..... ln x = u
=> e^(ln ln ln ln ..... ln x) = e^u
=> u=e^u
=> u * e^(-u) = 1
=> -u * e^(-u) = -1
=> LambertW(-u * e^(-u)) = LambertW(-1)
=> -u=LambertW(-1)
=> u = -LambertW(-1) which is equal to the constant you have found
The limit is always the same, regardless the initial x you may choose!!!!!
sorry for the late reply. Great explanation. Thanks!
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