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Complex variables for fractional-order systems

Emmanuel A. Gonzalez

Abstract

This paper discusses and summarizes some results on complex variables that are very useful in fractional-order systems analysis and design, specifically when the system is analyzed in the frequency domain. The author hopes that this document will serve as a handy reference when performing computations with complex variables, especially when working within the Laplace and Fourier domains. The reader can refer to Table I for the summary of these formulas.

Complex variables for fractional-order systems

Abstract

This paper discusses and summarizes some results on complex variables that are very useful in fractional-order systems analysis and design, specifically when the system is analyzed in the frequency domain. The author hopes that this document will serve as a handy reference when performing computations with complex variables, especially when working within the Laplace and Fourier domains. The reader can refer to Table I for the summary of these formulas.

Paper Structure

This paper contains 7 sections, 22 equations, 1 table.

Table of Contents

  1. Basics
  2. Roots of Complex Variables
  3. Results
  4. Preliminary Case: $s^{\alpha}=(\sigma+j\omega)^{\alpha}$ where $0<\alpha<1$ and $\sigma,\omega>0$
  5. Case 1: $s^{\alpha}=(j\omega)^{\alpha}$ where $0<\alpha<1$ and $\omega>0$
  6. Case 2: $a s^{\alpha}+b = a(j\omega)^{\alpha}+b$ where $0<\alpha<1$, $\omega>0$, and $a,b>0$
  7. Summary