Partial fractions

Level UniversityDifficulty ★★★★★Method⌖ Open in the map

What is it?

Split a rational function into simple pieces $\frac{A}{x - r}$ , $\frac{Bx + C}{x^2 + px + q}$ that integrate to logarithms and arctangents. Engineers use the same decomposition to invert Laplace and Z transforms.

Formulas

\frac{1}{x^2 - 1} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1}

Where it shows up in computing

Control theory★★★★★frequentRobotics and control
The time response of a system is found by splitting $G(s)$ into partial fractions and inverting each term.
Digital filters★★★★★frequentSignals, media and vision
Inverse Z-transforms and parallel filter realizations use partial fractions of $H(z)$ .

Where is it used?

Computing topics reachable from here, through the chain of ideas that leads to them:

⚙ Robotics and control

Control theory★★★★★
Control theory→Trajectory optimization and MPC★★★★★

∿ Signals, media and vision

Digital filters★★★★★

This page has the essentials. A fuller treatment (intuition, formal definition, worked example) is on the way.