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This work extends Euler's identity to reveal a unified framework for four fundamental dynamical classes, suggesting a deeper connection among them.
Euler’s identity eiθ = cos θ + i sin θ expresses rotation in the complex plane. We extend this identity by replacing the imaginary unit i with an operator J satisfying J^2 = −I, where J acts on a doubled dynamical state encoding the four fundamental PDE universality classes: elliptic, hyperbolic, parabolic (diffusive), and unitary (Schrodinger-type). This gives the Generalized Euler Identity eθJ = cos θ I + sin θ J , which performs Wick rotation through physical law itself. This formalism reveals a natural holofractal cyclicity among the canonical dynamical regimes.
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Nick Navid Yazdani (2026) studied this question.
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