05/22/2023, 03:46 PM
I am working on using the exponential function's fixed point at \(\infty\) in an attempt to define tetration \(^z a\) for \(a>e^{1/e}\). Generalizing \(e^{-\frac{1}{z}}\) to \(g(z)=1/f{(\frac{1}{z})}\) where \(f(\infty)=\infty\).
The iterates of \(g(z)\) are,
\[\frac{1}{f\left(\frac{1}{z}\right)},\frac{1}{f\left(f\left(\frac{1}{z}\right)\right)},\frac{1}{f\left(f\left(f\left(\frac{1}{z}\right)\right)\right)},\frac{1}{f\left(f\left(f\left(f\left(\frac{1}{z}\right)\right)\right)\right)}\]
The iterates of \(g(z)\) are,
\[\frac{1}{f\left(\frac{1}{z}\right)},\frac{1}{f\left(f\left(\frac{1}{z}\right)\right)},\frac{1}{f\left(f\left(f\left(\frac{1}{z}\right)\right)\right)},\frac{1}{f\left(f\left(f\left(f\left(\frac{1}{z}\right)\right)\right)\right)}\]
Daniel