# 2: Complex Numbers and Arithmetic, Laplace Transforms, and Partial-Fraction Expansion

- Page ID
- 7634

- 2.1: Review of Complex Numbers and Arithmetic
- We will find many uses in system dynamics for analysis with complex numbers and variables.

- 2.2: Introduction to Application of Laplace Transforms
- The Laplace transform (after French mathematician and celestial mechanician Pierre Simon Laplace, 1749-1827) is a mathematical tool primarily for solving ODEs, but with other important applications in system dynamics that we will study later.

- 2.3: Partial-Fraction Expansion
- Let us examine in more detail the justification for the form of the partial-fraction expansion. A ratio of polynomials, in which the numerator has a lower degree than that of the denominator, can usually be expanded into the simple partial-fraction form.

*Thumbnail: A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram representing the complex plane. "Re" is the real axis, "Im" is the imaginary axis, and i satisfies \(i^2 = −1\). (CC BY-SA 3.0 unported; Wolfkeeper via Wikipedia)*