| viXra.org > Functions and Analysis > viXra:2610.0017 |
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| viXra.org > Functions and Analysis > viXra:2610.0017 |
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Functions and Analysis |
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Authors: Brian Scannell
The Lorentz factor γ(β) =1/ √︁(1 — β^2), where β = v/c, is normally considered for real |β| < 1. Allowing its argument to range over the complex plane gives Γ(z) =1/√(1 — z^2), whose real and imaginary parts generate a striking system of level curves organized around the branch points z = +/-1. The Cauchy—Riemann equations explain why the regular curves from the two families intersect orthogonally. More surprisingly, eliminating the square root shows that each nonzero contour lies on a degree-eight algebraic locus. This apparent complexity is deceptive. Solving instead for z gives z = +/-√︁ (1 — Γ^-2), revealing that the apparently complicated contours in the z-plane are locally the preimages of vertical and horizontal straight lines in the Γ-plane. Thus, a familiar real function from special relativity provides an elementary example in which complicated algebraic contours conceal a simple conformal geometry.
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[v1] 2026-10-04 22:46:29
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