Trigonometric integrals

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

What is it?

Integrals of products of $\sin$ and $\cos$ . The key fact: different frequencies are orthogonal — $\int_0^{2\pi}\sin(mx)\sin(nx)\,\dd x = 0$ for $m \ne n$ — which is what makes Fourier series work.

Formulas

\int_0^{2\pi}\sin(mx)\sin(nx)\,\dd x = \pi\,\delta_{mn}, \qquad \int_0^{2\pi}\sin(mx)\cos(nx)\,\dd x = 0

Where it shows up in computing

Signal processing★★★★★frequentSignals, media and vision
Orthogonality lets a filter bank or a DFT pick out one frequency and ignore the rest.

Where is it used?

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

What depends on it

Fourier series

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

What is it?

Formulas

Where it shows up in computing

Where is it used?

⚛ Physics and simulation

∿ Signals, media and vision

⊕ Cryptography and security

ψ Quantum computing and physics

ℒ AI and machine learning

What depends on it