Complex Frequency Analysis from the Perspective of the Seven-Space Iterative System and Fourier Analysis: The Case a = 0.5

Starting from the derivative invariant u^(n) = (-π/s)^n u of the seven-space iterative system and the derivative relation d^n/dx^n e^(iωx) = (iω)^n e^(iωx) in Fourier analysis, a transformation relation iω = -π/s is obtained. For an arbitrary complex number s = a + ib, the complex frequency is derived as: ω = iπ/s = πb/(a²+b²) + i πa/(a²+b²) where the real part corresponds to the oscillation frequency and the imaginary part corresponds to the decay rate. This paper analyzes the distribution of complex frequencies on the line a = 0.5. Numerical results show that on this particular line, the decay rate decreases as the frequency increases, exhibiting an inverse relationship. This provides a concrete example of the connection between the seven-space system and Fourier analysis.