I introduce the Fourier transform as a generalization of the Fourier series from a finite interval to the whole real line. I give two basic examples that start to show the mathematics underlying Heisenberg’s uncertainty principle in quantum mechanics.
Video. Notes.

Last part! I give a few more details on how to find the radial part of the eigenfunctions of the Laplacian with vanishing boundary conditions on a sphere. This radial part is given by the spherical Bessel functions and their roots, and the latter quantize the eigenvalues of the full problem. A brief example is given at the end, but if you want to understand more, remember to play with the solutions!
Video. Notes.

More details on solving the diffusion equation in finite interval in 1D. Generalization to 3D in cartesian and spherical coordinates. Further details on the latter in part 3!
Video. Notes.

Generalized Fourier series. A few comments and two examples: spherical harmonics and Chebyshev polynomials. An application to PDEs: diffusion in a finite interval in one spatial dimension. More in part 2!

Video. Notes.

Using the binomial theorem, I derive the Fourier series for [Sin(x)]^n in exponential form, without doing any integrals.

Video. Notes.

From the trigonometric (sine/cosine) form to the complex exponential form of the Fourier series, using Euler’s identity.
Video. Notes.