$\varrho$: The Self-Referential Fixed Point of the Complex Exponential

Abstract

I was taught that $e^x=x$ has no solution, and taught to leave it at that. But in mathematics "no solution" has usually meant "not on this line yet": $x^2=-1$ waited for the complex plane, and $e^x=x$ turns out to be waiting there too. Over $\mathbb{C}$ the exponential has a fixed point $\varrho =0.318\dots +1.337\dots i$, the unique solution of $\exp (z)=z$ in the strip $0<\mathrm{Im}z<\pi$ (equivalently $-W_{-1}(-1)$), and it carries more structure than its one-line definition lets on. At $\varrho$ the rectangular and log-polar coordinates of a complex number coincide, forcing the identities $\mathrm{Re}\varrho =\log |\varrho |$ and $\arg \varrho =\mathrm{Im}\varrho$. As a dynamical point $\varrho$ is repelling for $\exp$ and attracting for $\log$, linearizable for both by one Koenigs coordinate, and the base of a transpose identity $w^{\varrho }={\varrho }^{\log w}$. It generates an aperiodic log-polar lattice and sits a hair off a clean relation with $\pi$, namely $\mathrm{Re}(\varrho )\pi =0.99944\dots$. Passing to the octonions, the fixed points of ${\exp }_{\mathbb{O}}$ fill concentric six-spheres, the innermost $\mathrm{Re}(\varrho )+\mathrm{Im}(\varrho )S^6$, whose triples obey an exact identity $I_4^2+\frac{1}{4}{I}_5^2=\mathrm{Gram}$ carrying one invariant, a twist angle, absent from ordinary spherical trigonometry. Throughout, what is proved is kept apart from what is only computed.

12 pages, 2 figures