Continuing logarithm \(\displaystyle{\log{{\left({\log{{\left(\ldots{\log{{\left({z}\right)}}}\right)}}}\right)}}}\)

Guadalupe Glass

Guadalupe Glass

Answered question

2022-03-27

Continuing logarithm log(log(log(z)))

Answer & Explanation

Esteban Sloan

Esteban Sloan

Beginner2022-03-28Added 21 answers

If it converges, then
a=log(log(log(z)))=log(a)
Thus,
a=log(a)
ea=a1=aeaa=Wk(1)k=00.3181.337i
is the Lambert W function. Note that this is constant, and dependent only on whether or not the choice of z converge and where z is. The position of z will determine which branch of the Lambert W function it will converge to. A couple notes:
If we z is a perfect super power of e, then it diverges due to log(0). That is, zeee and z0.
I am pretty sure it converges everywhere else, with the exception of
z=W(1).
Not finished with the rest:

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