Classical to Quantum | Intro to Harmonic Analysis on a Circle (Rationally!) | Wild Egg Maths

The harmonic analysis of spheres, particularly the circle S^1, the 2 dimensional sphere S^2, and the 3 dimensional sphere S^3, figure importantly in QM. The usual story for the circle is Fourier series, where functions of an angle theta parametrizing the circle are expanded into sums of cosine and sine functions of multiples of that angle. However this requires both the transcendental notion of an "angle" and the transcendental circular functions. In the spirit of Rational Trigonometry, we look for an alternative description which stays inside the elementary domain of algebra, polynomials and there ongoing extensions power series. But we will use quite different language: we prefer to work with polynumbers, which are arrays of coefficients, and incorporate some aspects of the very new "Box Arithmetic" developed in the Math Foundations series at Insights into Mathematics to deal with bi-polynumbers and Derivatives of them in a systematic way. The Laplacian and the concept of harmonic polynumbers are introduced used novel factorial notation which makes understanding them quite a lot simpler. By the end of the lecture we will have the essentials for a purely rational form of Fourier series, and will be well prepared for the exciting addition of complex number algebra into the story in the next video in the series! This is part of the Playlist Classical To Quantum, where we look at numerous issues in the borderlands of physics and mathematics, but from a mathematician's point of view. Many thanks to all Members of the Channel, and to Patreon supporters. My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/... My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things. Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... If you would like to support these new initiatives for mathematics education and research, you could also consider become a Patreon supporter. Thank you. Here are the Wild Egg Maths Playlists (some available only to Members!)    • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist   Here are the Insights into Mathematics Playlists:    • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist      • Playlist   *****************