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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2606.01668 |
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| _version_ | 1866916072238612480 |
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| author | Bickford, Michael |
| author_facet | Bickford, Michael |
| contents | 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 $\varrho = 0.318\ldots + 1.337\ldots\,i$, the unique solution of $\exp(z) = z$ in the strip $0 < \operatorname{Im} z < π$ (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 $\operatorname{Re}\varrho = \log|\varrho|$ and $\arg\varrho = \operatorname{Im}\varrho$. 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 $π$, namely $\operatorname{Re}(\varrho)\,π= 0.99944\ldots$. Passing to the octonions, the fixed points of $\exp_{\mathbb{O}}$ fill concentric six-spheres, the innermost $\operatorname{Re}(\varrho) + \operatorname{Im}(\varrho)\,S^6$, whose triples obey an exact identity $I_4^2 + \tfrac14 I_5^2 = \operatorname{Gram}$ carrying one invariant, a twist angle, absent from ordinary spherical trigonometry. Throughout, what is proved is kept apart from what is only computed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01668 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $\varrho$: The Self-Referential Fixed Point of the Complex Exponential Bickford, Michael Complex Variables 30D05 (Primary), 33B10, 37F10, 17A35, 11J81 (Secondary) 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 $\varrho = 0.318\ldots + 1.337\ldots\,i$, the unique solution of $\exp(z) = z$ in the strip $0 < \operatorname{Im} z < π$ (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 $\operatorname{Re}\varrho = \log|\varrho|$ and $\arg\varrho = \operatorname{Im}\varrho$. 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 $π$, namely $\operatorname{Re}(\varrho)\,π= 0.99944\ldots$. Passing to the octonions, the fixed points of $\exp_{\mathbb{O}}$ fill concentric six-spheres, the innermost $\operatorname{Re}(\varrho) + \operatorname{Im}(\varrho)\,S^6$, whose triples obey an exact identity $I_4^2 + \tfrac14 I_5^2 = \operatorname{Gram}$ carrying one invariant, a twist angle, absent from ordinary spherical trigonometry. Throughout, what is proved is kept apart from what is only computed. |
| title | $\varrho$: The Self-Referential Fixed Point of the Complex Exponential |
| topic | Complex Variables 30D05 (Primary), 33B10, 37F10, 17A35, 11J81 (Secondary) |
| url | https://arxiv.org/abs/2606.01668 |