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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.

Posted byjoaquindrutDecember 8, 2022December 8, 2022Posted inFourier series and transforms

Fourier series: example 1

A first example of calculating a Fourier series explicitly. More examples in upcoming videos!

Video. Notes.

Posted byjoaquindrutDecember 7, 2022Posted inFourier series and transforms

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.

Posted byjoaquindrutDecember 6, 2022Posted inComplex numbers, Fourier series and transforms, Linear algebra, Quantum mechanics

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.

Posted byjoaquindrutDecember 5, 2022Posted inFourier series and transforms

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.

Posted byjoaquindrutDecember 2, 2022December 2, 2022Posted inComplex numbers

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.

Posted byjoaquindrutDecember 1, 2022Posted inComplex numbers

More about Gaussian integrals

I show how to calculate a Gaussian integral in arbitrary dimensions in the presence of a linear source term, as a first step toward applications in quantum field theory and statistical mechanics.

Video. Notes.

Posted byjoaquindrutNovember 30, 2022November 30, 2022Posted inOther fun stuff

The adjoint operator

A quick look at the adjoint of a linear operator (aka the hermitian conjugate). An example with a differential operator and how to prove properties using the definition based on the inner product.

Video. Notes.

Posted byjoaquindrutNovember 29, 2022November 29, 2022Posted inLinear algebra

A 1d Green’s function example two ways

A closer look at how to calculate a Green’s function in a 1D example, using a direct solution and a spectral representation.

Video. Notes.

Posted byjoaquindrutNovember 28, 2022November 28, 2022Posted inGreen's functions, Linear algebra, Other fun stuff

Diagonalizing differential operators on the lattice

How do you represent a function on a computer with discrete memory? What about its derivatives? A first look at stuff on a lattice, the matrix form of the first-derivative operator (forward, with periodic boundary conditions), and its eigenvectors and eigenvalues (Fourier modes).

Video. Notes.

Posted byjoaquindrutNovember 25, 2022Posted inFourier series and transforms, Linear algebra

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