The best articles by Ciro Santilli Updated 2025-07-16
These are the best articles ever authored by Ciro Santilli, most of them in the format of Stack Overflow answers.
Ciro posts update about new articles on his Twitter accounts.
A chronological list of all articles is also kept at: Section "Updates".
Some random generally less technical in-tree essays will be present at: Section "Essays by Ciro Santilli".
Computational physics Updated 2025-07-16
The intersection of two beautiful arts: coding and physics!
Computational physics is a good way to get valuable intuition about the key equations of physics, and train your numerical analysis skills:
Fourier transform Updated 2025-07-16
Continuous version of the Fourier series.
Of course, every function defined on a finite line segment (i.e. a compact space).
Therefore, the Fourier transform can be seen as a generalization of the Fourier series that can also decompose functions defined on the entire real line.
As a more concrete example, just like the Fourier series is how you solve the heat equation on a line segment with Dirichlet boundary conditions as shown at: Section "Solving partial differential equations with the Fourier series", the Fourier transform is what you need to solve the problem when the domain is the entire real line.
History of the Fourier series Updated 2025-07-16
First published by Fourier in 1807 to solve the heat equation.
Laplace's equation Updated 2025-07-16
Like a heat equation but for functions without time dependence, space-only.
TODO confirm: does the solution of the heat equation always converge to the solution of the Laplace equation as time tends to infinity?
In one dimension, the Laplace equation is boring as it is just a straight line since the second derivative must be 0. That also matches our intuition of the limit solution of the heat equation.
Robin boundary condition Updated 2025-07-16
Examples: