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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