Functions and Analysis

    

A Two-Phase Extension of Euler's Formula in an Idempotent Representation

Authors: Gorchilin Vyacheslav

We consider a two-phase representation of Euler's formula in the commutative algebra A ≃ ℂ × ℂ. An idempotent decomposition and explicitly selected logarithm values define continuous real powers of a hyperbolic unit j and of −j. Their product gives a family J(a,b) with two independent phases. We derive its exponential representation, two-phase sine and cosine, the fundamental trigonometric identity, and the composition law. The phase parameter plane, the torus of values, and the linear multiplication action are distinguished. The construction uses established bicomplex analysis and presents a single sequence from selected continuous powers to two independent rotations.

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[v1] 2026-11-14 09:01:25

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