A quick reminder of the derivation of the geometric sum and its limit as a series. Video. Notes.
Monthly Archives: December 2022
The binomial theorem
A simple proof, by induction, of the binomial theorem. Video. Notes.
Fourier series: [sin(x)]^n
Using the binomial theorem, I derive the Fourier series for [Sin(x)]^n in exponential form, without doing any integrals. Video. Notes.
Fourier series: exponential form
From the trigonometric (sine/cosine) form to the complex exponential form of the Fourier series, using Euler’s identity.Video. Notes.
Fourier series: example 2
A quick discussion of exp(-x^2) cos(3x), followed by an explicit derivation of the Fourier series of sin^3(x) without doing integrals.Video. Notes.
Fourier series: example 1
A first example of calculating a Fourier series explicitly. More examples in upcoming videos! Video. Notes.
The quantum free particle
Using background on complex numbers and Fourier modes, here is a first discussion of the most essential quantum mechanical system. Video. Notes.
Fourier series: first concepts
A first look at the definition and concept of Fourier series as a global expansion in a finite interval, compared with a Taylor series as a local expansion. More in upcoming videos! Video. Notes.
Euler’s formula
A quick look at the derivation of Euler’s formula and its uses and connections to trigonometric and hyperbolic sine and cosine functions of complex variables. Video. Notes.
Complex numbers basics
Definition and basic properties of complex numbers. Addition, multiplication, division, conjugation. Polar representation. Trigonometric identities. Differentiation & Cauchy-Riemann conditions. Simple examples. Video. Notes.