The ultimate beta method

[tommy1729]

You can also see Kneser as the unique function \(f\) such that:

\[
|f'(z+1)| =|e^{f(z)}|\\
\]

if we assume that \(|f'(z)| =1\).........

This is explained through basic level set calculus....

What ?!?

That seems like a universal tetration property for complex continu ( not even analytic required ) solutions.

f(z+1) = exp(f(z))

so by the chain rule

f ' (z+1) = exp'(f(z)) f'(z) = exp(f(z)) * a = exp(f(z) + ln(a))

so if f ' (z) = a and a is a root of unity :

| f ' (z+1) | = | exp'(f(z)) f'(z) | = | exp(f(z)) * a | = | exp(f(z) + ln(a)) | = |a| | exp(f(z)) | = | exp(f(z)) |

QED

All we need is some vague notion of differentiable to define some at least a weak derivative consistant over the domains and ranges considered.



regards

tommy1729