欧拉复数表达式与恒等式的修正——基于旋转-缩放正交解耦的几何重建与离散数值验证Correction of Euler's Complex Expression and Identity: A Geometric Reconstruction Based on Rotation-Scaling Orthogonal Decoupling with Discrete Numerical Validation

⚠️ 勘误与纠正声明(版本更新) 本人早几天的欧拉修正存在严重错误。现在统一纠正如下: 修正核心为 a^{ σ+iθ} =a ^σ ⋅(cosθ+i⋅sinθ),其中 σ控制缩放、θ控制旋转,旋转角度与底数无关。 在这个体系下,建议实部和虚部都严格遵循不省略的表达。否则会发生逻辑混乱。 省略书写是造成混乱的原因之一。 实部虚部零值和不操作关系: • σ=0,θ ≠0:伸缩归一的旋转。任何非零模长都归一的旋转。 • σ ≠0,θ=0:只伸缩不旋转(模长 a σ ,相位不变)。 • σ=1,θ=0:不伸缩也不旋转(模长保持底数 a),基准状态。 • σ=0,θ=0:伸缩归一且不旋转(模长归 1,结果为原相位)。 经典 e^{ iθ }代表纯旋转,其实是对 σ=1的省略书写,即纯旋转。但是被误解为 σ=0的省略表达。(2026.08.27) 本文基于旋转-缩放正交解耦的几何原理,对欧拉复数表达式与欧拉恒等式提出修正。经典表达式 e^ (iθ) =cosθ+isinθ修正为 e ^(iθ) =e(cosθ+isinθ),等价形式为 1^ iθ =cosθ+isinθ;对应恒等式修正为 e^ iπ +e=0。给出了修正体系的完整公理与运算法则(乘法、除法、共轭、整数幂、微分、积分)。在七个基准问题(简谐振动、浅水方程、刚性ODE、二维涡流、洛伦兹系统、热传导、非线性薛定谔方程)上的数值实验表明,结合投影算子后,修正体系消除了守恒量的长期漂移。作者声明无高等数学科班教育背景,本工作基于朴素几何直觉与逻辑一致性审视,AI辅助完成推演与数值实现。 This preprint proposes a correction to Euler's formula and Euler's identity based on the geometric principle of rotation-scaling orthogonal decoupling. The classical expression e iθ =cosθ+isinθis revised to e iθ =e(cosθ+isinθ), with an equivalent form 1 iθ =cosθ+isinθ. The corresponding Euler's identity becomes e iπ +e=0. A complete set of axioms and operational rules for the revised system is presented, including multiplication, division, conjugation, integer powers, differentiation, and integration. Numerical experiments on seven benchmark problems (harmonic oscillator, shallow water equations, stiff ODEs, Navier-Stokes vortex, Lorenz system, heat equation, and nonlinear Schrödinger equation) demonstrate that the revised framework eliminates long-term drift in conserved quantities when combined with a projection operator. The author declares no formal mathematical training; this work is driven by geometric intuition and logical consistency checking with AI assistance.